<PAGE>   1
                                                                    EXHIBIT 4.13


                            FORM OF 3-3/4% DEBENTURE

THE HOLDER OF THIS SECURITY BY ITS ACCEPTANCE HEREOF IS DEEMED TO HAVE AGREED TO
BE BOUND BY THE PROVISIONS OF A REGISTRATION RIGHTS AGREEMENT BETWEEN LIBERTY
AND THE INITIAL PURCHASER NAMED THEREIN, DATED FEBRUARY 10, 2000 (THE
"REGISTRATION RIGHTS AGREEMENT"). LIBERTY WILL PROVIDE A COPY OF THE
REGISTRATION RIGHTS AGREEMENT TO SUCH HOLDER WITHOUT CHARGE UPON WRITTEN REQUEST
TO LIBERTY AT ITS PRINCIPAL PLACE OF BUSINESS.

THE SECURITY EVIDENCED HEREBY HAS NOT BEEN REGISTERED UNDER THE SECURITIES ACT
OF 1933, AS AMENDED (THE "SECURITIES ACT"), OR ANY STATE OR OTHER SECURITIES
LAWS. NEITHER THIS SECURITY NOR ANY INTEREST OR PARTICIPATION HEREIN MAY BE
REOFFERED, SOLD, ASSIGNED, TRANSFERRED, PLEDGED, ENCUMBERED OR OTHERWISE
DISPOSED OF IN THE ABSENCE OF SUCH REGISTRATION OR UNLESS THE TRANSACTION IS
EXEMPT FROM, OR NOT SUBJECT TO, THE REGISTRATION REQUIREMENTS OF THE SECURITIES
ACT. THE HOLDER OF THIS SECURITY BY ITS ACCEPTANCE HEREOF (1) REPRESENTS THAT
(A) IT IS A "QUALIFIED INSTITUTIONAL BUYER" (AS DEFINED IN RULE 144A UNDER THE
SECURITIES ACT), (B) IT IS AN INSTITUTIONAL "ACCREDITED INVESTOR" WITHIN THE
MEANING OF SUBPARAGRAPH (a)(1), (2), (3) OR (7) OF RULE 501 UNDER THE SECURITIES
ACT OR (C) IT IS NOT A U.S. PERSON AND IS ACQUIRING THIS SECURITY IN AN
"OFFSHORE TRANSACTION" PURSUANT TO RULE 903 OR 904 OF REGULATION S, (2) AGREES
THAT IT WILL NOT PRIOR TO (X) THE DATE WHICH IS TWO YEARS (OR SUCH SHORTER
PERIOD OF TIME AS PERMITTED BY RULE 144(k) UNDER THE SECURITIES ACT OR ANY
SUCCESSOR PROVISION THEREUNDER) AFTER THE LATER OF THE ORIGINAL ISSUE DATE
HEREOF (OR OF ANY PREDECESSOR OF THIS SECURITY) OR THE LAST DAY ON WHICH LIBERTY
OR ANY AFFILIATE OF LIBERTY WAS THE OWNER OF THIS SECURITY (OR ANY PREDECESSOR
OF THIS SECURITY) AND (Y) SUCH LATER DATE, IF ANY, AS MAY BE REQUIRED BY
APPLICABLE LAWS (THE "RESALE RESTRICTION TERMINATION DATE"), OFFER, SELL OR
OTHERWISE TRANSFER THIS SECURITY EXCEPT (A) TO LIBERTY OR ANY SUBSIDIARY
THEREOF, (B) PURSUANT TO A REGISTRATION STATEMENT WHICH HAS BEEN DECLARED
EFFECTIVE UNDER THE SECURITIES ACT, (C) FOR SO LONG AS THE SECURITIES ARE
ELIGIBLE FOR RESALE PURSUANT TO RULE 144A, TO A PERSON IT REASONABLY BELIEVES IS
A "QUALIFIED INSTITUTIONAL BUYER" AS DEFINED IN RULE 144A UNDER THE SECURITIES
ACT THAT PURCHASES FOR ITS OWN ACCOUNT OR FOR THE ACCOUNT OF A QUALIFIED
INSTITUTIONAL BUYER TO WHOM NOTICE IS GIVEN THAT THE TRANSFER IS BEING MADE IN
RELIANCE ON RULE 144A, (D) INSIDE THE UNITED STATES TO AN INSTITUTIONAL
"ACCREDITED INVESTOR" WITHIN THE MEANING OF SUBPARAGRAPH (a)(1), (2), (3) OR (7)
OF RULE 501 UNDER THE SECURITIES ACT THAT PRIOR TO SUCH TRANSFER, FURNISHES (OR
HAS FURNISHED ON ITS BEHALF BY A U.S. BROKER-DEALER) TO THE TRUSTEE OR TRANSFER
AGENT, A SIGNED LETTER CONTAINING CERTAIN REPRESENTATIONS AND AGREEMENTS
RELATING TO THE RESTRICTIONS ON TRANSFER OF THIS SECURITY (THE FORM OF WHICH
LETTER CAN BE OBTAINED FROM THE TRUSTEE OR TRANSFER AGENT FOR THIS SECURITY),
(E) PURSUANT TO OFFERS AND SALES TO NON-U.S. PERSONS THAT OCCUR OUTSIDE THE
UNITED STATES WITHIN THE MEANING OF REGULATION S UNDER THE SECURITIES ACT,
PURSUANT TO RULE 904 OF REGULATION S OR (F) PURSUANT TO ANOTHER AVAILABLE
EXEMPTION FROM THE REGISTRATION REQUIREMENTS OF THE SECURITIES ACT, AND (3)
AGREES THAT IT WILL GIVE TO EACH PERSON TO WHOM THIS SECURITY IS TRANSFERRED A
NOTICE SUBSTANTIALLY TO THE EFFECT OF THIS LEGEND; PROVIDED THAT LIBERTY, THE



<PAGE>   2


TRUSTEE AND THE TRANSFER AGENT AND REGISTRAR RESERVE THE RIGHT PRIOR TO ANY
OFFER, SALE OR OTHER TRANSFER PURSUANT TO CLAUSES (D), (E) OR (F) ABOVE TO
REQUIRE DELIVERY OF AN OPINION OF COUNSEL, CERTIFICATIONS AND OTHER INFORMATION
SATISFACTORY TO LIBERTY, THE TRUSTEE AND THE TRANSFER AGENT AND REGISTRAR. THIS
LEGEND WILL BE REMOVED UPON THE REQUEST OF THE HOLDER AFTER THE RESALE
RESTRICTION TERMINATION DATE. AS USED HEREIN, THE TERMS "OFFSHORE TRANSACTION,"
"UNITED STATES" AND "U.S. PERSON" HAVE THE RESPECTIVE MEANINGS GIVEN TO THEM BY
REGULATION S UNDER THE SECURITIES ACT. IN ANY CASE, ANY HOLDER HEREOF THAT HAS
ACQUIRED THIS SECURITY IN AN "OFFSHORE TRANSACTION" PURSUANT TO RULE 903 OR 904
OF REGULATION S WILL NOT, DIRECTLY OR INDIRECTLY, ENGAGE IN ANY HEDGING
TRANSACTION WITH REGARD TO THIS SECURITY OR ANY SECURITY ISSUABLE UPON EXCHANGE
OF THIS SECURITY EXCEPT AS PERMITTED BY THE SECURITIES ACT.

UNLESS THIS SECURITY IS PRESENTED BY AN AUTHORIZED REPRESENTATIVE OF THE
DEPOSITORY TRUST COMPANY, A NEW YORK CORPORATION ("DTC"), TO LIBERTY OR ITS
AGENT FOR REGISTRATION OF TRANSFER, EXCHANGE, OR PAYMENT, AND ANY CERTIFICATE
ISSUED IS REGISTERED IN THE NAME OF CEDE & CO. OR IN SUCH OTHER NAME AS IS
REQUESTED BY AN AUTHORIZED REPRESENTATIVE OF DTC (AND ANY PAYMENT IS MADE TO
CEDE & CO. OR TO SUCH OTHER ENTITY AS IS REQUESTED BY AN AUTHORIZED
REPRESENTATIVE OF DTC), ANY TRANSFER, PLEDGE, OR OTHER USE HEREOF FOR VALUE OR
OTHERWISE BY OR TO ANY PERSON IS WRONGFUL, INASMUCH AS THE REGISTERED OWNER
HEREOF, CEDE & CO., HAS AN INTEREST HEREIN.

No. R-1                                                             $750,000,000
CUSIP No. 530715AK7

                            LIBERTY MEDIA CORPORATION

                 3 3/4% Senior Exchangeable Debentures due 2030

                               Rule 144A Debenture

         Liberty Media Corporation, a Delaware corporation (hereinafter called
the "Company", which term includes any successor corporation under the Indenture
referred to below), for value received, hereby promises to pay to Cede & Co., or
registered assigns, the amount provided in Section 204 of the Fourth
Supplemental Indenture referred to herein (such amount being referred to herein
as the Maturity Repayment Amount) on February 15, 2030, and to pay interest on
the Original Principal Amount of this Debenture from February 10, 2000, or from
the most recent date to which interest has been paid or provided for,
semiannually on February 15 and August 15 in each year (each, an "Interest
Payment Date"), commencing August 15, 2000, at the rate of 3 3/4% per annum,
until the Maturity Repayment Amount is paid or made available for payment.
Interest on this Debenture shall be calculated on the basis of a 360-day year
consisting of twelve 30-day months. The interest so payable and paid or provided
for on any Interest Payment Date will, as provided in such Indenture, be paid to
the Person in whose name this Debenture (or one or more Predecessor Securities)
is registered at the close of business on the Regular Record Date for such
interest, which shall be the February 1 or August 1 (whether or not a Business
Day), as



                                       2
<PAGE>   3


the case may be, immediately preceding such Interest Payment Date. Any such
interest which is payable, but is not paid or provided for, on any Interest
Payment Date shall forthwith cease to be payable to the registered Holder hereof
on the relevant Regular Record Date by virtue of having been such Holder, and
may be paid to the Person in whose name this Debenture (or one or more
Predecessor Securities) is registered at the close of business on a Special
Record Date for the payment of such Defaulted Interest to be fixed by the
Company, notice whereof shall be given to the Holders of Debentures not less
than 10 days prior to such Special Record Date, or may be paid at any time in
any other lawful manner not inconsistent with the requirements of any securities
exchange on which the Debentures may be listed, and upon such notice as may be
required by such exchange, all as more fully provided in such Indenture.

         Payment of the Maturity Repayment Amount and the interest on this
Debenture will be made at the office or agency of the Company maintained for
that purpose in The Borough of Manhattan, The City of New York, in such coin or
currency of the United States of America as at the time of payment is legal
tender for payment of public and private debts; provided, however, that, at the
option of the Company, interest may be paid by check mailed to the address of
the Person entitled thereto as such address shall appear in the Security
Register; provided, further, that payment to DTC or any successor Depository may
be made by wire transfer to the account designated by DTC or such successor
Depository in writing.

         This Security is a permanent Rule 144A Debenture issued on the date
hereof which represents $750,000,000 of the Original Principal Amount of the
Company's 3 3/4% Senior Exchangeable Debentures due 2030, offered and sold to
qualified institutional buyers, as defined in Rule 144A under the Securities
Act. This Debenture is one of a duly authorized issue of securities of the
Company (herein called the "Debentures") issued and to be issued in one or more
series under an Indenture dated as of July 7, 1999 (herein called, together with
the Fourth Supplemental Indenture referred to below and all other indentures
supplemental thereto, the "Indenture") between the Company and The Bank of New
York, as Trustee (herein called the "Trustee", which term includes any successor
trustee under the Indenture), to which Indenture and all indentures supplemental
thereto reference is hereby made for a statement of the respective rights,
limitations of rights, duties and immunities thereunder of the Company, the
Trustee and the Holders of the Debentures, and of the terms upon which the
Debentures are, and are to be, authenticated and delivered. This Debenture is
one of the series designated on the face hereof, initially limited (subject to
exceptions provided in the Indenture) to the aggregate Original Principal Amount
specified in the Fourth Supplemental Indenture between the Company and the
Trustee, dated as of February 10, 2000, establishing the terms of the Debentures
pursuant to the Indenture (the "Fourth Supplemental Indenture").

         The Debentures are redeemable at the option of the Company, in whole or
in part at any time or from time to time on or after February 15, 2004, on the
terms set forth in Section 208(a) of the Fourth Supplemental Indenture. In
addition, the Debentures are redeemable at the option of the Company upon the
occurrence of a Tax Event or a Share Event or following an Excess Borrow Cost
Period, each as defined in the Fourth Supplemental Indenture, on the terms set
forth in Sections 208(b), (c) and (d), respectively, of the Fourth Supplemental
Indenture.

         The Debentures are exchangeable at the option of the Holders thereof,
on the terms set forth in Section 209 of the Fourth Supplemental Indenture.



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         If an Event of Default (as defined in the Indenture, including the
amendments thereto in the Fourth Supplemental Indenture) with respect to the
Debentures shall occur and be continuing, the principal of the Debentures may be
declared due and payable in the manner and with the effect provided in the
Indenture.

         The Indenture permits, with certain exceptions as therein provided, the
amendment thereof and the modification of the rights and obligations of the
Company and the rights of the Holders of the Securities of each series issued
under the Indenture at any time by the Company and the Trustee with the consent
of the Holders of not less than a majority in aggregate principal amount of the
Securities at the time Outstanding of each series affected thereby. The
Indenture also contains provisions permitting the Holders of specified
percentages in aggregate principal amount of the Securities of any series at the
time Outstanding, on behalf of the Holders of all Securities of such series, to
waive compliance by the Company with certain provisions of the Indenture and
certain past defaults under the Indenture and their consequences. Any such
consent or waiver by the Holder of this Debenture shall be conclusive and
binding upon such Holder and upon all future Holders of this Debenture and of
any Debentures issued upon the registration of transfer hereof or in exchange
herefor or in lieu hereof, whether or not notation of such consent or waiver is
made upon this Debenture or such Debentures.

         No reference herein to the Indenture and no provision of this Debenture
or of the Indenture shall alter or impair the obligation of the Company, which
is absolute and unconditional, to pay the Maturity Repayment Amount and interest
on this Debenture, at the times, place and rate, and in the coin or currency,
herein and in the Indenture prescribed.

         As provided in the Indenture and subject to certain limitations set
forth therein and in this Debenture, the transfer of this Debenture may be
registered on the Security Register upon surrender of this Debenture for
registration of transfer at the office or agency of the Company maintained for
the purpose in any place where the Maturity Repayment Amount and interest on
this Debenture are payable, duly endorsed by, or accompanied by a written
instrument of transfer in form satisfactory to the Company and the Security
Registrar duly executed by, the Holder hereof or by his attorney duly authorized
in writing, and thereupon one or more new Debentures of this series and of like
tenor, of authorized denominations and for the same aggregate Original Principal
Amount, will be issued to the designated transferee or transferees.

         The Debentures are issuable only in registered form without coupons in
the denominations specified in the Fourth Supplemental Indenture establishing
the terms of the Debentures, all as more fully provided in the Indenture. As
provided in the Indenture, and subject to certain limitations set forth in the
Indenture and in this Debenture, the Debentures are exchangeable for a like
aggregate Original Principal Amount of Debentures of this series in different
authorized denominations, as requested by the Holders surrendering the same.

         No service charge shall be made for any such registration of transfer
or exchange, but the Company may require payment of a sum sufficient to cover
any tax or other governmental charge payable in connection therewith, other than
in certain cases provided in the Indenture.

         Prior to due presentment of this Debenture for registration of
transfer, the Company, the Trustee and any agent of the Company or the Trustee
may treat the Person in whose name this



                                       4
<PAGE>   5


Debenture is registered as the owner hereof for all purposes, whether or not
this Debenture be overdue, and neither the Company, the Trustee nor any such
agent shall be affected by notice to the contrary.

         This Debenture shall be governed by and construed in accordance with
the laws of the State of New York.

         All terms used in this Debenture which are defined in the Indenture
shall have the meanings assigned to them in the Indenture.

         Unless the certificate of authentication hereon has been executed by or
on behalf of the Trustee under the Indenture by the manual signature of one of
its authorized signatories, this Debenture shall not be entitled to any benefits
under the Indenture or be valid or obligatory for any purpose.



                                       5
<PAGE>   6


                                                                    EXHIBIT 4.13


                            FORM OF 3-3/4% DEBENTURE

         IN WITNESS WHEREOF, the Company has caused this instrument to be duly
executed.




                                                   LIBERTY MEDIA CORPORATION



Attest:                                            By:
        -----------------------                       --------------------------
        Name:                                         Name:
        Title:                                        Title:



                     TRUSTEE'S CERTIFICATE OF AUTHENTICATION

         This is one of the Debentures of the series designated herein referred
to in the within-mentioned Indenture.

Dated: February 10, 2000                           THE BANK OF NEW YORK,
                                                   as Trustee


                                                   By:
                                                      --------------------------
                                                          Authorized Signatory





                                       6
<PAGE>   7



                             CERTIFICATE OF TRANSFER


         To transfer or assign this Debenture, fill in the form below:

I or we transfer and assign this Debenture to


--------------------------------------------------------------------------------
                       (Insert assignee's tax I.D. number)



--------------------------------------------------------------------------------
--------------------------------------------------------------------------------
--------------------------------------------------------------------------------
              (Print or Type assignee's name, address and zip code)

and irrevocably appoint ________________ agent to transfer this Debenture on the
books of the Company. The agent may substitute another to act for him.


Date:                                    Your signature:
     -------------------                                ------------------------





                                       7
<PAGE>   8







                                   SCHEDULE A

                              SCHEDULE OF EXCHANGES

The following exchanges of Debentures represented by this Rule 144A Debenture
have been made:

<TABLE>
<CAPTION>

                                                 Change in Original      Original Principal
Original Principal                               Principal Amount of     Amount of this Rule
Amount of this Rule                              this Rule 144A          144A Debenture
144A Debenture as of      Date exchange          Debenture due to        following such
February 10, 2000         Made                   Exchange                exchange               Notation made by
------------------------- ---------------------- ----------------------- ---------------------- ----------------------
<S>                       <C>                    <C>                     <C>                    <C>
      $750,000,000
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========================= ====================== ======================= ====================== ======================
</TABLE>





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