<PAGE>

                                                                   EXHIBIT 10.12

                              AMENDMENT NO. 1 TO
                             INVESTMENT AGREEMENT
                          (Series E Preferred Stock)

     This Amendment No. 1 to Investment Agreement (the "Amendment") dated as of
                                                        ---------
January 14, 2000 (the "Effective Date"), is entered into by and among Greenwich
                       --------------
Technology Partners, Inc., a Delaware corporation (the "Corporation") and those
                                                        -----------
parties executing the signature page hereto, representing at least 66 2/3% of
the Corporation's outstanding Series E Preferred Stock, $0.01 par value per
share (the "Series E Shares").
            ---------------

     This Amendment amends that certain Investment Agreement (Series E Preferred
Stock) dated September 10, 1999 (the "Original Agreement").
                                      ------------------

     Capitalized terms used herein and not otherwise defined, shall have the
meanings set forth in the Original Agreement.

     NOW, THEREFORE, in accordance with Section 10.2 of the Original Agreement,
the parties hereto agree that the Original Agreement shall be amended as
follows:

     1.   Section 2.2(b) shall be deleted in its entirety and replaced with the
following:

          "(b)  One or more additional closings (each an "Additional Closing")
                                                          ------------------
          of the sale and purchase of up to an additional 4,265,402 of the
          Series E Shares under this Agreement shall take place at the offices
          of Testa, Hurwitz & Thibeault, LLP, 125 High Street, Boston,
          Massachusetts, 02110, at 1:00 p.m. on such date or dates as is
          mutually agreeable to the Additional Investors participating in such
          Additional Closing, and the Company."

     2.   Section 2.2(c) shall be deleted in its entirety and replaced with the
following:

          "(c)  The Initial Closing and the Additional Closings are each
          referred to herein as a "Closing"."
                                   -------

     3.   Section 2.5 shall be deleted in its entirety and replaced with the
following:

          "2.5  Counterpart Signature Pages. At each Additional Closing, the
          Additional Investors participating in such Additional Closing shall
          execute a counterpart signature page to this Agreement, the
          Shareholders' Agreement and the Registration Rights Agreement."

     4.   Exhibit A, Schedule of Investors shall be deleted in its entirety and
          ---------
replaced with the attached Exhibit A hereto.
                           ---------

     5.   All other provisions of the Original Agreement shall be unmodified and
shall remain in full force and effect.
<PAGE>

     6.   This Amendment may be signed in one or more counterparts, each of
which shall be deemed to be an original and all of which shall be considered one
and the same agreement and shall become effective when each of the parties has
signed and delivered a counterpart to the other.

     7.   This Amendment shall be governed by and interpreted in accordance with
the laws of the State of Delaware, but without giving effect to any Delaware
choice of law provisions which might otherwise make the laws of a difference
jurisdiction govern or apply.

                 [Remainder of Page Left Blank Intentionally]
<PAGE>

   IN WITNESS WHEREOF, this Amendment has been executed as of the date and year
first above written.

                                    CORPORATION:

                                    GREENWICH TECHNOLOGY PARTNERS, INC.

                                    By: /s/ Dennis M. Goett
                                       ---------------------------------------
                                    Name:   Dennis M. Goett
                                    Title: CFO

                                    INVESTORS:

                                    CHASE VENTURE CAPITAL ASSOCIATES, L.P.

                                    By:

                                    By: /s/ Michael R. Hannon
                                       ---------------------------------------
                                    Name:   Michael R. Hannon
                                    Title: General Partner

                                    CREDIT SUISSE FIRST BOSTON VENTURE FUND I,
                                    L.P.

                                    By: QBB Management I, LLC, Its General
                                        Partner

                                    By: /s/ George Bourtos
                                       ---------------------------------------
                                    Name:   George Bourtos
                                    Title: Member

                                    STV PARTNERS III, L.L.C.

                                    By:

                                    By: /s/ Jerome C. Silvey
                                       ---------------------------------------
                                    Name:   Jerome C. Silvey
                                    Title: Managing Member
<PAGE>

                                    FG-GTPF

                                    By:

                                    By: /s/ Kathleen E. Shepphird
                                       ----------------------------------------
                                    Name:   Kathleen E. Shepphird
                                    Title: Managing Director

                                    VANTAGEPOINT COMMUNICATIONS PARTNERS, LP

                                    By:

                                    By: /s/ William Jefferson Marshall
                                       ----------------------------------------
                                    Name:   William Jefferson Marshall
                                    Title: Managing Director

                                    VANTAGEPOINT VENTURE PARTNERS 1996, L.P.

                                    By: Vantagepoint Communications Associates,
                                        LLC, Its General Partner

                                    By: /s/ William Jefferson Marshall
                                       ----------------------------------------
                                    Name:   William Jefferson Marshall
                                    Title: Managing Director

                                    WHEATLEY PARTNERS II L.P.

                                    By:

                                    By: /s/ Irwin Lieber
                                       ---------------------------------------
                                    Name:   Irwin Lieber
                                    Title: General Partner
<PAGE>

                                   EXHIBIT A
                                   ---------

                             Schedule of Investors
                             ---------------------

                                Initial Closing
                                ---------------

<TABLE>
<CAPTION>
                                                                                Aggregate
                                                        Number of Series E    Purchase Price
                                                           Shares to be        for Series E
Name of Series E Investor                                    Purchased            Shares
-------------------------                                    ---------            ------
<S>                                                     <C>                   <C>
Chase Venture Capital Associates, L.P.                       4,739,337        $10,000,001.07
Credit Suisse First Boston Venture Fund I, L.P.                236,967        $   500,000.37
STV Partners III, L.L.C.                                       473,934        $ 1,000,000.74
FG-GTPF                                                        236,967        $   500,000.37
Vantagepoint Communications Partners, LP                       947,867        $ 1,999,999.37
Vantagepoint Venture Partners 1996, L.P.                       473,934        $ 1,000,000.74
Wheatley Partners II L.P.                                      473,934        $ 1,000,000.74

                                    TOTAL:                   7,582,940        $16,000,003.40
</TABLE>

                          First Additional Closing
                          ------------------------

<TABLE>
<CAPTION>
                                                                                Aggregate
                                                        Number of Series E    Purchase Price
                                                           Shares to be        for Series E
Name of Series E Investor                                    Purchased            Shares
-------------------------                                    ---------            ------
<S>                                                     <C>                   <C>
Edwin J. O'Mara                                                  7,108             14,997.88
John Stopper                                                    23,696             49,998.56
Laurence Pinkus                                                  4,739              9,999.29
Kevin Bock                                                      50,236            105,997.96
James Tucci                                                     59,241            124,998.51
Robert Berlin                                                   11,848             24,999.28
John V. Wheeler                                                 11,848             24,999.28
L. David Cardenas and Stacey J. Cardenas                         2,369              4,998.59
Louis J. Mischianti                                             23,696             49,998.56
James A. Conroy                                                  4,739              9,999.29
Emil A. Roymans                                                  2,369              4,998.59
Paul A. Rubin                                                    7,109             14,999.99
Scott M. Freeman                                                 4,739              9,999.29
Michael J. Schmidtberger                                         4,739              9,999.29
Michael A. Grossman                                              5,924             12,499.64
Karl Frey                                                       11,848             24,999.28
EPFL Partners                                                    4,739              9,999.29
John D. Miller                                                  11,848             24,999.28
</TABLE>
<PAGE>

Deborah A. Farrington                                    2,369        4,998.59
Gerard F. Becker and Christine B. Becker                11,848       24,999.28
Nils A. Johnson and Patricia A. Johnson                  9,478       19,998.58
Clemente Family Trust                                    9,478       19,998.58
BMZ Investments                                         23,696       49,998.56
Richard J. Testa                                         9,952       20,998.72
Anthony M. Carvette                                     23,696       49,998.56
Greg Berger                                              5,924       12,499.64
Kevin J. Kitson                                         47,393       99,999.23
Brian J. Flynn                                           4,739        9,999.29
Angus M. Green                                          18,957       39,999.27
Joseph A. Cabrera                                       11,848       24,999.28
Clint Heiden                                            23,696       49,998.56
Stephen B. Seigel                                       47,393       99,999.23
David HW Turner                                         11,848       24,999.28
Dennis M. Goett                                         18,957       39,999.27
IRA FBO Dennis M. Goett                                 28,435       59,997.85
S. Elizabeth Press and Mark Andrew Mohn                 10,000          21,100
Paul C. Carey                                           18,957       39,999.27
Samer Tawfik                                           142,180      299,999.80
Highwood Partners LLC                                   14,218       29,999.98
William C. Cox III and Beatrice I. Cox                   9,478       19,998.58
Gregory W. Carney                                       11,848       24,999.28
Paul T. Goldman                                          7,109       14,999.99
Donald K. Bryan and Belinda B. Bryan                    11,848       24,999.28
Edward Cettina                                           7,109       14,999.99
Elio Cettina                                             7,109       14,999.99
The CIT Group/Equity Investments, Inc.                 473,934    1,000,000.74
Chase Venture Capital Associates, L.P.                 473,933      999,998.63
VantagePoint Venture Partners 1996, L.P.               157,820      333,000.20
VantagePoint Communications Partners, L.P.             316,114      667,000.54
Total:                                               2,224,199   $4,693,059.89
<PAGE>

                           Second Additional Closing
                           -------------------------

<TABLE>
<CAPTION>
                                                                                    Aggregate
                                                           Number of Series E     Purchase Price
                                                              Shares to be         for Series E
         Name of Series E Investor                              Purchased             Shares
         -------------------------                              ---------             ------
<S>                                                        <C>                    <C>
Mark B. Templeton                                                 11,848             24,999.28
Mark Wolfenberger                                                 23,696             49,998.56
Frank P. Slattery                                                  4,739              9,999.29
Jeffrey Wrona                                                      4,739              9,999.29
Michael Ma                                                         4,739              9,999.29
Kurt Weber                                                        11,848             24,999.28
UGE Enterprises LLC                                                4,739              9,999.29
Edgell Street Partners                                             7,109             14,999.99
Dennis M. Goett                                                   14,218             29,999.98
Jill Catania                                                       4,739              9,999.29
Geryl Darington                                                   11,848             24,999.28
Robert Garbarino                                                  23,696             49,998.56
Donald Henderson                                                   9,478             19,998.58
Bruce M. Tanis                                                    11,848             24,999.28
Peter Cherasia                                                    23,696             49,998.56
Michael Tunstall                                                  35,545             74,999.95
Wayne A. Segal                                                    23,696             49,998.56
Anthony Trousett                                                  23,696             49,998.56
S. Elizabeth Press and Mark Andrew Mohn                           20,000             42,200.00
Richard Haverly                                                  100,000            211,000.00
Kevin Barry                                                        4,739              9,999.29
High Street Investors 2000                                        47,393             99,999.23
TOTAL:                                                           428,049           $903,183.39
</TABLE>