<PAGE>
                                  EXHIBIT 3.2
 
                             ROSLYN BANCORP, INC.

                          AMENDED AND RESTATED BYLAWS

                           ARTICLE I - STOCKHOLDERS


    Section 1.     Annual Meeting.
    ---------      -------------- 

    An annual meeting of the stockholders, for the election of Directors to
succeed those whose terms expire and for the transaction of such other business
as may properly come before the meeting, shall be held at such place, on such
date, and at such time as the Board of Directors shall each year fix, which date
shall be within thirteen (13) months subsequent to the later of the date of
incorporation or the last annual meeting of stockholders.

    Section 2.     Special Meetings.
    ---------      ---------------- 

    Subject to the rights of the holders of any class or series of preferred
stock of the Corporation, special meetings of stockholders of the Corporation
may be called only by the Board of Directors pursuant to a resolution adopted by
a majority of the total number of Directors which the Corporation would have if
there were no vacancies on the Board of Directors (hereinafter the "Whole
Board").

    Section 3.     Notice of Meetings.
    ---------      ------------------ 

    Written notice of the place, date, and time of all meetings of the
stockholders shall be given, not less than ten (10) nor more than sixty (60)
days before the date on which the meeting is to be held, to each stockholder
entitled to vote at such meeting, except as otherwise provided herein or
required by law (meaning, here and hereinafter, as required from time to time by
the Delaware General Corporation Law or the Certificate of Incorporation of the
Corporation).

    When a meeting is adjourned to another place, date or time, written notice
need not be given of the adjourned meeting if the place, date and time thereof
are announced at the meeting at which the adjournment is taken; provided,
however, that if the date of any adjourned meeting is more than thirty (30) days
after the date for which the meeting was originally noticed, or if a new record
date is fixed for the adjourned meeting, written notice of the place, date, and
time of the adjourned meeting shall be given in conformity herewith.  At any
adjourned meeting, any business may be transacted which might have been
transacted at the original meeting.

    Section 4.     Quorum.
    ---------      ------ 

    At any meeting of the stockholders, the holders of a majority of all of the
shares of the stock entitled to vote at the meeting, present in person or by
proxy (after giving effect to the provisions of Article FOURTH of the
Corporation's Certificate of Incorporation), shall constitute a quorum for all
purposes, unless or except to the extent that the presence of a larger number
may be required by 
<PAGE>
 
law. Where a separate vote by a class or classes is required, a majority of the
shares of such class or classes present in person or represented by proxy (after
giving effect to the provisions of Article FOURTH of the Corporation's
Certificate of Incorporation) shall constitute a quorum entitled to take action
with respect to that vote on that matter.

    If a quorum shall fail to attend any meeting, the chairman of the meeting or
the holders of a majority of the shares of stock entitled to vote who are
present, in person or by proxy, may adjourn the meeting to another place, date,
or time.

    If a notice of any adjourned special meeting of stockholders is sent to all
stockholders entitled to vote thereat, stating that it will be held with those
present in person or by proxy constituting a quorum, then except as otherwise
required by law, those present in person or by proxy at such adjourned meeting
shall constitute a quorum, and all matters shall be determined by a majority of
the votes cast at such meeting.

    Section 5.     Organization.
    ---------      ------------ 

    Such person as the Board of Directors may have designated or, in the absence
of such a person, the Chairman of the Board of the Corporation or, in his or her
absence, such person as may be chosen by the holders of a majority of the shares
entitled to vote who are present, in person or by proxy, shall call to order any
meeting of the stockholders and act as chairman of the meeting.  In the absence
of the Secretary of the Corporation, the secretary of the meeting shall be such
person as the chairman appoints.

    Section 6.     Conduct of Business.
    ---------      ------------------- 

         (a)  The chairman of any meeting of stockholders shall determine the
order of business and the procedures at the meeting, including such regulation
of  the manner of voting and the conduct of discussion as seem to him or her in
order.  The date and time of the opening and closing of the polls for each
matter upon which the stockholders will vote at the meeting shall be announced
at the meeting.

         (b)  At any annual meeting of the stockholders, only such business
shall be conducted as shall have been brought before the meeting:  (i) by or at
the direction of the Board of Directors or (ii) by any stockholder of the
Corporation who is entitled to vote with respect thereto and who complies with
the notice procedures set forth in this Section 6(b).  For business to be
properly brought before an annual meeting by a stockholder, the business must
relate to a proper subject matter for stockholder action and the stockholder
must have given timely notice thereof in writing to the Secretary of the
Corporation.  To be timely, a stockholder's notice must be delivered or mailed
to and received at the principal executive offices of the Corporation not less
than ninety (90) days prior to the date of the annual meeting; provided,
however, that in the event that less than one hundred (100) days' notice or
prior public disclosure of the date of the meeting is given or made to
stockholders, notice by the stockholder to be timely must be received not later
than the close of business on the 10th day following the day on which such
notice of the date of the annual meeting was mailed or such public disclosure
was made.  A stockholder's notice to the Secretary shall set 

                                       2
<PAGE>
 
forth as to each matter such stockholder proposes to bring before the annual
meeting: (i) a brief description of the business desired to be brought before
the annual meeting and the reasons for conducting such business at the annual
meeting; (ii) the name and address, as they appear on the Corporation's books,
of the stockholder proposing such business; (iii) the class and number of shares
of the Corporation's capital stock that are beneficially owned by such
stockholder; and (iv) any material interest of such stockholder in such
business. Notwithstanding anything in these Bylaws to the contrary, no business
shall be brought before or conducted at an annual meeting except in accordance
with the provisions of this Section 6(b). The Officer of the Corporation or
other person presiding over the annual meeting shall, if the facts so warrant,
determine and declare to the meeting that business was not properly brought
before the meeting in accordance with the provisions of this Section 6(b) and,
if he should so determine, he shall so declare to the meeting and any such
business so determined to be not properly brought before the meeting shall not
be transacted.

    At any special meeting of the stockholders, only such business shall be
conducted as shall have been brought before the meeting by or at the direction
of the Board of Directors.

         (c)  Only persons who are nominated in accordance with the procedures
set forth in these Bylaws shall be eligible for election as Directors.
Nominations of persons for election to the Board of Directors of the Corporation
may be made at a meeting of stockholders at which directors are to be elected
only:  (i) by or at the direction of the Board of Directors; or (ii) by any
stockholder of the Corporation entitled to vote for the election of Directors at
the meeting who complies with the notice procedures set forth in this Section
6(c).  Such nominations, other than those made by or at the direction of the
Board of Directors, shall be made by timely notice in writing to the Secretary
of the Corporation.  To be timely, a stockholder's notice shall be delivered or
mailed to and received at the principal executive offices of the Corporation not
less than ninety (90) days prior to the date of the meeting; provided, however,
that in the event that less than one hundred (100) days' notice or prior
disclosure of the date of the meeting is given or made to stockholders, notice
by the stockholder to be timely must be so received not later than the close of
business on the 10th day following the day on which such notice of the date of
the meeting was mailed or such public disclosure was made. Such stockholder's
notice shall set forth:  (i) as to each person whom such stockholder proposes to
nominate for election or re-election as a Director, all information relating to
such person that is required to be disclosed in solicitations of proxies for
election of directors, or is otherwise required, in each case pursuant to
Regulation 14A under the Securities Exchange Act of 1934, as amended (including
such person's written consent to being named in the proxy statement as a nominee
and to serving as a director if elected); and (ii) as to the stockholder giving
the notice (x) the name and address, as they appear on the Corporation's books,
of such stockholder and (y) the class and number of shares of the Corporation's
capital stock that are beneficially owned by such stockholder.  At the request
of the Board of Directors, any person nominated by the Board of Directors for
election as a Director shall furnish to the Secretary of the Corporation that
information required to be set forth in a stockholder's notice of nomination
which pertains to the nominee.  No person shall be eligible for election as a
Director of the Corporation unless nominated in accordance with the provisions
of this Section 6(c).  The Officer of the Corporation or other person presiding
at the meeting shall, if the facts so warrant, determine that a nomination was
not made in accordance with such provisions and, if he or she shall so
determine, he or she shall so declare to the meeting and the defective
nomination shall be disregarded.

                                       3
<PAGE>
 
    Section 7.     Proxies and Voting.
    ---------      ------------------ 

    At any meeting of the stockholders, every stockholder entitled to vote may
vote in person or by proxy authorized by an instrument in writing filed in
accordance with the procedure established for the meeting.  Any facsimile
telecommunication or other reliable reproduction of the writing or transmission
created pursuant to this paragraph may be substituted or used in lieu of the
original writing or transmission for any and all purposes for which the original
writing or transmission could be used, provided that such copy, facsimile
telecommunication or other reproduction shall be a complete reproduction of the
entire original writing or transmission.

    All voting, including on the election of Directors but excepting where
otherwise required by law or by the governing documents of the Corporation, may
be made by a voice vote; provided, however, that upon demand therefor by a
stockholder entitled to vote or his or her proxy, a stock vote shall be taken.
Every stock vote shall be taken by ballot, each of which shall state the name of
the stockholder or proxy voting and such other information as may be required
under the procedures established for the meeting.  The Corporation shall, in
advance of any meeting of stockholders, appoint one or more inspectors to act at
the meeting and make a written report thereof.  The Corporation may designate
one or more persons as alternate inspectors to replace any inspector who fails
to act.  If no inspector or alternate is able to act at a meeting of
stockholders, the person presiding at the meeting shall appoint one or more
inspectors to act at the meeting.  Each inspector, before entering upon the
discharge of his duties, shall take and sign an oath faithfully to execute the
duties of inspector with strict impartiality and according to the best of his
ability.

    All elections shall be determined by a plurality of the votes cast, and
except as otherwise required by law or the Certificate of Incorporation, all
other matters shall be determined by a majority of the votes cast.

    Section 8.     Stock List.
    ---------      ---------- 

    A complete list of stockholders entitled to vote at any meeting of
stockholders, arranged in alphabetical order for each class of stock and showing
the address of each such stockholder and the number of shares registered in his
or her name, shall be open to the examination of any such stockholder, for any
purpose germane to the meeting, during ordinary business hours for a period of
at least ten (10) days prior to the meeting, either at a place within the city
where the meeting is to be held, which place shall be specified in the notice of
the meeting, or if not so specified, at the place where the meeting is to be
held.


    The stock list shall also be kept at the place of the meeting during the
whole time thereof and shall be open to the examination of any such stockholder
who is present.  This list shall presumptively determine the identity of the
stockholders entitled to vote at the meeting and the number of shares held by
each of them.

    Section 9.     Consent of Stockholders in Lieu of Meeting.
    ---------      ------------------------------------------ 

                                       4
<PAGE>
 
    Subject to the rights of the holders of any class or series of preferred
stock of the Corporation, any action required or permitted to be taken by the
stockholders of the Corporation must be effected at an annual or special meeting
of stockholders of the Corporation and may not be effected by any consent in
writing by such stockholders.


                        ARTICLE II - BOARD OF DIRECTORS

    Section 1.     General Powers, Number, Term of Office and Limitations.
    ---------      ------------------------------------------------------ 

    The business and affairs of the Corporation shall be under the direction of
its Board of Directors.  The number of Directors who shall constitute the Whole
Board shall be such number as the Board of Directors shall from time to time
have designated, except that in the absence of such designation shall be  ten.
The Board of Directors shall annually elect a Chairman of the Board from among
its members who shall, when present, preside at its meetings.

    Except for the initial members of the Board of Directors as of the effective
date of these Bylaws, no person shall be eligible for initial election as a
Director who is 68 years of age or more. No person may be elected, appointed,
nominated or otherwise serve as a Director of the Corporation after December 31
of the year in which such person attains the age of 75.  Vacancies on the Board
of Directors created by operation of this provision may be filled in accordance
with these Bylaws.

    The Directors, other than those who may be elected by the holders of any
class or series of Preferred Stock, shall be divided, with respect to the time
for which they severally hold office, into three classes, with the term of
office of the first class to expire at the first annual meeting of stockholders,
the term of office of the second class to expire at the annual meeting of
stockholders one year thereafter and the term of office of the third class to
expire at the annual meeting of stockholders two years thereafter, with each
Director to hold office until his or her successor shall have been duly elected
and qualified.  At each annual meeting of stockholders, Directors elected to
succeed those Directors whose terms then expire shall be elected for a term of
office to expire at the third succeeding annual meeting of stockholders after
their election, with each Director to hold office until his or her successor
shall have been duly elected and qualified.

                                       5
<PAGE>
 
    Section 2.     Vacancies and Newly Created Directorships.
    ---------      ----------------------------------------- 

    Subject to the rights of the holders of any class or series of Preferred
Stock, and unless the Board of Directors otherwise determines, newly created
directorships resulting from any increase in the authorized number of directors
or any vacancies in the Board of Directors resulting from death, resignation,
retirement, disqualification, removal from office or other cause may be filled
only by a majority vote of the Directors then in office, though less than a
quorum, and Directors so chosen shall hold office for a term expiring at the
annual meeting of stockholders at which the term of office of the class to which
they have been elected expires and until such Director's successor shall have
been duly elected and qualified.  No decrease in the number of authorized
directors constituting the Board shall shorten the term of any incumbent
Director.

    Section 3.     Regular Meetings.
    ---------      ---------------- 

    Regular meetings of the Board of Directors shall be held at such place or
places, on such date or dates, and at such time or times as shall have been
established by the Board of Directors and publicized among all Directors.  A
notice of each regular meeting shall not be required.

    Section 4.     Special Meetings.
    ---------      ---------------- 

    Special meetings of the Board of Directors may be called by one-third (1/3)
of the Directors then in office (rounded up to the nearest whole number), by the
Chairman of the Board or the President or, in the event that the Chairman of the
Board or President are incapacitated or otherwise unable to call such meeting,
by the Secretary, and shall be held at such place, on such date, and at such
time as they, or he or she, shall fix.  Notice of the place, date, and time of
each such special meeting shall be given each Director by whom it is not waived
by mailing written notice not less than five (5) days before the meeting or by
telegraphing or telexing or by facsimile transmission of the same not less than
twenty-four (24) hours before the meeting.  Unless otherwise indicated in the
notice thereof, any and all business may be transacted at a special meeting.

    Section 5.     Quorum.
    ---------      ------ 

    At any meeting of the Board of Directors, a majority of the Whole Board
shall constitute a quorum for all purposes.  If a quorum shall fail to attend
any meeting, a majority of those present may adjourn the meeting to another
place, date, or time, without further notice or waiver thereof.

    Section 6.     Participation in Meetings By Conference Telephone.
    ---------      ------------------------------------------------- 

    Members of the Board of Directors, or of any committee thereof, may
participate in a meeting of such Board or committee by means of conference
telephone or similar communications equipment by means of which all persons
participating in the meeting can hear each other and such participation shall
constitute presence in person at such meeting.

    Section 7.     Conduct of Business.
    ---------      ------------------- 

                                       6
<PAGE>
 
    At any meeting of the Board of Directors, business shall be transacted in
such order and manner as the Board may from time to time determine, and all
matters shall be determined by the vote of a majority of the Directors present,
except as otherwise provided herein or required by law. Action may be taken by
the Board of Directors without a meeting if all members thereof consent thereto
in writing, and the writing or writings are filed with the minutes of
proceedings of the Board of Directors.

    Section 8.     Powers.
    ---------      ------ 

    The Board of Directors may, except as otherwise required by law, exercise
all such powers and do all such acts and things as may be exercised or done by
the Corporation.

    Section 9.     Compensation of Directors.
    ---------      ------------------------- 

    Directors, as such, may receive, pursuant to resolution of the Board of
Directors, fixed fees and other compensation for their services as Directors,
including, without limitation, their services as members of committees of the
Board of Directors.


                           ARTICLE III - COMMITTEES

    Section 1.     Committees of the Board of Directors.
    ---------      ------------------------------------ 

    The Board of Directors, by a vote of a majority of the Board of Directors,
may from time to time designate committees of the Board, with such lawfully
delegable powers and duties as it thereby confers, to serve at the pleasure of
the Board and shall, for these committees and any others provided for herein,
elect a Director or Directors to serve as the member or members, designating, if
it desires, other Directors as alternate members who may replace any absent or
disqualified member at any meeting of the committee.  Any committee so
designated may exercise the power and authority of the Board of Directors to
declare a dividend, to authorize the issuance of stock or to adopt a certificate
of ownership and merger pursuant to Section 253 of the Delaware General
Corporation Law if the resolution which designates the committee or a
supplemental resolution of the Board of Directors shall so provide.  In the
absence or disqualification of any member of any committee and any alternate
member in his or her place, the member or members of the committee present at
the meeting and not disqualified from voting, whether or not he or she or they
constitute a quorum, may by unanimous vote appoint another member of the Board
of Directors to act at the meeting in the place of the absent or disqualified
member.

                                       7
<PAGE>
 
    Section 2.     Conduct of Business.
    ---------      ------------------- 

    Each committee may determine the procedural rules for meeting and conducting
its business and shall act in accordance therewith, except as otherwise provided
herein or required by law. Adequate provision shall be made for notice to
members of all meetings.  The quorum requirements for each such committee shall
be a majority of the members of such committee unless otherwise determined by
the Board of Directors by a majority vote of the Board of Directors which such
quorum determined by a majority of the Board may be one-third of such members
and all matters considered by such committees shall be determined by a majority
vote of the members present. Action may be taken by any committee without a
meeting if all members thereof consent thereto in writing, and the writing or
writings are filed with the minutes of the proceedings of such committee.

    Section 3.     Nominating Committee.
    ----------     -------------------- 

    The Board of Directors shall appoint a Nominating Committee of the Board,
consisting of not less than three (3) members.  The Nominating Committee shall
have authority:  (a) to review any nominations for election to the Board of
Directors made by a stockholder of the Corporation pursuant to Section 6(c)(ii)
of Article I of these Bylaws in order to determine compliance with such Bylaw;
and (b) to recommend to the Whole Board nominees for election to the Board of
Directors to replace those Directors whose terms expire at the annual meeting of
stockholders next ensuing.

                             ARTICLE IV - OFFICERS

    Section 1.     Generally.
    ---------      --------- 

         (a) The Board of Directors as soon as may be practicable after the
annual meeting of stockholders shall choose a Chairman of the Board, Chief
Executive Officer, a President, one or more Vice Presidents, a Secretary and a
Treasurer and from time to time may choose such other officers as it may deem
proper.  The Chairman of the Board shall be chosen from among the Directors.
Any number of offices may be held by the same person.

         (b) The term of office of all Officers shall be until the next annual
election of Officers and until their respective successors are chosen but any
Officer may be removed from office at any time by the affirmative vote of a
majority of the authorized number of Directors then constituting the Board of
Directors.

         (c) All Officers chosen by the Board of Directors shall have such
powers and duties as generally pertain to their respective Offices, subject to
the specific provisions of this ARTICLE IV.  Such officers shall also have such
powers and duties as from time to time may be conferred by the Board of
Directors or by any committee thereof.

         (d) The Board of Directors may, except as otherwise required by law,
remove any Officer of the Corporation with or without cause, and from time to
time, devolve the powers and duties of any Officer upon any other person for the
time being, and to confer upon any Officer of the Corporation the power to
appoint, remove or suspend subordinate officers, employees and agents.

                                       8
<PAGE>
 
    Section 2.     Chairman of the Board of Directors.
    ---------      ---------------------------------- 

    The Chairman of the Board shall, subject to the provisions of these Bylaws
and to the direction of the Board of Directors, serve in general executive
capacity and unless the Board has designated another person, when present, shall
preside at all meetings of the stockholders of the Corporation. The Chairman of
the Board shall perform all duties and have all powers which are commonly
incident to the office of Chairman of the Board or which are delegated to him or
her by the Board of Directors.  He or she shall have power to sign all stock
certificates, contracts and other instruments of the Corporation which are
authorized.

    Section 3.     President and Chief Executive Officer.
    ---------      ------------------------------------- 

    The President and Chief Executive Officer (the "President") shall have
general responsibility for the management and control of the business and
affairs of the Corporation and shall perform all duties and have all powers
which are commonly incident to the offices of President and Chief Executive
Officer or which are delegated to him or her by the Board of Directors.  Subject
to the direction of the Board of Directors, the President shall have power to
sign all stock certificates, contracts and other instruments of the Corporation
which are authorized and shall have general supervision of all of the other
Officers (other than the Chairman of the Board), employees and agents of the
Corporation.

    Section 4.     Vice President.
    ----------     -------------- 
 
    The Vice President or Vice Presidents shall perform the duties of the
President in his absence or during his inability to act.  In addition, the Vice
Presidents shall perform the duties and exercise the powers usually incident to
their respective offices and/or such other duties and powers as may be properly
assigned to them by the Board of Directors, the Chairman of the Board or the
President. A Vice President or Vice Presidents may be designated as Executive
Vice President or Senior Vice President.

    Section 5.     Secretary.
    ---------      --------- 

    The Secretary or Assistant Secretary shall issue notices of meetings, shall
keep their minutes, shall have charge of the seal and the corporate books, shall
perform such other duties and exercise such other powers as are usually incident
to such office and/or such other duties and powers as are properly assigned
thereto by the Board of Directors, the Chairman of the Board or the President.
Subject to the direction of the Board of Directors, the Secretary shall have the
power to sign all stock certificates.

                                       9
<PAGE>
 
    Section 6.     Treasurer.
    ----------     ----------

    The Treasurer shall be the Comptroller of the Corporation and shall have the
responsibility for maintaining the financial records of the Corporation.  He or
she shall make such disbursements of the funds of the Corporation as are
authorized and shall render from time to time an account of all such
transactions and of the financial condition of the Corporation.  The Treasurer
shall also perform such other duties as the Board of Directors may from time to
time prescribe.  Subject to the direction of the Board of Directors, the
Treasurer shall have the power to sign all stock certificates.

    Section 7.     Assistant Secretaries and Other Officers.
    ---------      -----------------------------------------

    The Board of Directors may appoint one or more Assistant Secretaries and
such other Officers who shall have such powers and shall perform such duties as
are provided in these Bylaws or as may be assigned to them by the Board of
Directors, the Chairman of the Board or the President.

    Section 8.     Action with Respect to Securities of Other Corporations.
    ----------     --------------------------------------------------------

    Unless otherwise directed by the Board of Directors, the President or any
Officer of the Corporation authorized by the President shall have power to vote
and otherwise act on behalf of the Corporation, in person or by proxy, at any
meeting of stockholders of or with respect to any action of stockholders of any
other corporation in which this Corporation may hold securities and otherwise to
exercise any and all rights and powers which this Corporation may possess by
reason of its ownership of securities in such other corporation.

                               ARTICLE V - STOCK

    Section 1.     Certificates of Stock.
    ---------      --------------------- 

    Each stockholder shall be entitled to a certificate signed by, or in the
name of the Corporation by, the Chairman of the Board or the President, and by
the Secretary or an Assistant Secretary, or any Treasurer or Assistant
Treasurer, certifying the number of shares owned by him or her.  Any or all of
the signatures on the certificate may be by facsimile.

    Section 2.     Transfers of Stock.
    ---------      ------------------ 

    Transfers of stock shall be made only upon the transfer books of the
Corporation kept at an office of the Corporation or by transfer agents
designated to transfer shares of the stock of the Corporation.  Except where a
certificate is issued in accordance with Section 4 of Article V of these Bylaws,
an outstanding certificate for the number of shares involved shall be
surrendered for cancellation before a new certificate is issued therefor.

                                       10
<PAGE>
 
    Section 3.     Record Date.
    ---------      ----------- 

    In order that the Corporation may determine the stockholders entitled to
notice of or to vote at any meeting of stockholders, or to receive payment of
any dividend or other distribution or allotment of any rights or to exercise any
rights in respect of any change, conversion or exchange of stock or for the
purpose of any other lawful action, the Board of Directors may fix a record
date, which record date shall not precede the date on which the resolution
fixing the record date is adopted and which record date shall not be more than
sixty (60) nor less than ten (10) days before the date of any meeting of
stockholders, nor more than sixty (60) days prior to the time for such other
action as hereinbefore described; provided, however, that if no record date is
fixed by the Board of Directors, the record date for determining stockholders
entitled to notice of or to vote at a meeting of stockholders shall be at the
close of business on the day next preceding the day on which notice is given or,
if notice is waived, at the close of business on the next day preceding the day
on which the meeting is held, and, for determining stockholders entitled to
receive payment of any dividend or other distribution or allotment or rights or
to exercise any rights of change, conversion or exchange of stock or for any
other purpose, the record date shall be at the close of business on the day on
which the Board of Directors adopts a resolution relating thereto.

    A determination of stockholders of record entitled to notice of or to vote
at a meeting of stockholders shall apply to any adjournment of the meeting;
provided, however, that the Board of Directors may fix a new record date for the
adjourned meeting.

    Section 4.     Lost, Stolen or Destroyed Certificates.
    ---------      -------------------------------------- 

    In the event of the loss, theft or destruction of any certificate of stock,
another may be issued in its place pursuant to such regulations as the Board of
Directors may establish concerning proof of such loss, theft or destruction and
concerning the giving of a satisfactory bond or bonds of indemnity.

    Section 5.     Regulations.
    ---------      ----------- 

    The issue, transfer, conversion and registration of certificates of stock
shall be governed by such other regulations as the Board of Directors may
establish.


                              ARTICLE VI - NOTICES

    Section 1.     Notices.
    ---------      ------- 

    Except as otherwise specifically provided herein or required by law, all
notices required to be given to any stockholder, Director, Officer, employee or
agent shall be in writing and may in every instance be effectively given by hand
delivery to the recipient thereof, by depositing such notice in the mails,
postage paid, or by sending such notice by prepaid telegram or mailgram or other
courier. Any such notice shall be addressed to such stockholder, Director,
Officer, employee or agent at his or her last known address as the same appears
on the books of the Corporation.  The time when such 

                                       11
<PAGE>
 
notice is received, if hand delivered, or dispatched, if delivered through the
mails or by telegram or mailgram or other courier, shall be the time of the
giving of the notice.

    Section 2.     Waivers.
    ---------      ------- 

    A written waiver of any notice, signed by a stockholder, Director, Officer,
employee or agent, whether before or after the time of the event for which
notice is to be given, shall be deemed equivalent to the notice required to be
given to such stockholder, Director, Officer, employee or agent.  Neither the
business nor the purpose of any meeting need be specified in such a waiver.


                          ARTICLE VII - MISCELLANEOUS

    Section 1.     Facsimile Signatures.
    ---------      -------------------- 

    In addition to the provisions for use of facsimile signatures elsewhere
specifically authorized in these Bylaws, facsimile signatures of any officer or
officers of the Corporation may be used whenever and as authorized by the Board
of Directors or a committee thereof.

    Section 2.     Corporate Seal.
    ---------      -------------- 

    The Board of Directors may provide a suitable seal, containing the name of
the Corporation, which seal shall be in the charge of the Secretary.  If and
when so directed by the Board of Directors or a committee thereof, duplicates of
the seal may be kept and used by the Treasurer or by an Assistant Secretary or
an assistant to the Treasurer.

    Section 3.     Reliance Upon Books, Reports and Records.
    ---------      ---------------------------------------- 

    Each Director, each member of any committee designated by the Board of
Directors, and each Officer of the Corporation shall, in the performance of his
or her duties, be fully protected in relying in good faith upon the books of
account or other records of the Corporation and upon such information, opinions,
reports or statements presented to the Corporation by any of its Officers or
employees, or committees of the Board of Directors so designated, or by any
other person as to matters which such Director or committee member reasonably
believes are within such other person's professional or expert competence and
who has been selected with reasonable care by or on behalf of the Corporation.

    Section 4.     Fiscal Year.
    ---------      ----------- 

    The fiscal year of the Corporation shall be as fixed by the Board of
Directors.

    Section 5.     Time Periods.
    ---------      ------------ 

    In applying any provision of these Bylaws which requires that an act be done
or not be done a specified number of days prior to an event or that an act be
done during a period of a specified 

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number of days prior to an event, calendar days shall be used, the day of the
doing of the act shall be excluded, and the day of the event shall be included.

    Section 6.     Adoption of Regulations.
    ----------     ------------------------

    The Board of Directors may, except as otherwise required by law, adopt from
time to time regulations, not inconsistent with these Bylaws, for the management
of the Corporation's business and affairs.


                           ARTICLE VIII - AMENDMENTS

    The Board of Directors may amend, alter or repeal these Bylaws at any
meeting of the Board, provided notice of the proposed change was given not less
than two (2) days prior to the meeting. The stockholders shall also have power
to amend, alter or repeal these Bylaws at any meeting of stockholders provided
notice of the proposed change was given in the notice of the meeting; provided,
however, that, notwithstanding any other provisions of the Bylaws or any
provision of law which might otherwise permit a lesser vote or no vote, but in
addition to any affirmative vote of the holders of any particular class or
series of the voting stock required by law, the Certificate of Incorporation,
any Preferred Stock Designation or these Bylaws, the affirmative votes of the
holders of at least 80% of the voting power of all the then-outstanding shares
of the Voting Stock, voting together as a single class, shall be required to
alter, amend or repeal any provisions of these Bylaws.

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<PAGE>
 
    The above Bylaws are effective as of May 11, 1998, the date of incorporation
of Roslyn Bancorp, Inc.


                                 /s/ Mary M. Ehrich
                                ----------------------------------
                                Mary M. Ehrich
                                Secretary

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