Tables of approximate values of the first zeros on
the critical line of some primitive Dirichlet L-series

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Introduction Tables References Links Contact [Up]

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Introduction

This page presents the results of my efforts to compute the first zeros, on the critical line, of some Dirichlet L-series. Only some L-series associated with Dirichlet characters [1] were considered. The Dirichlet L-series associated with the simplest character is the well-known zeta function. Riemann, in a path-breaking paper [2], conjectured that the non-trivial zeros of this function have real part equal to 1/2 (the critical line). This constitutes the famous Riemann Hypothesis (RH). The Extended Riemann Hypothesis (ERH) asserts the same for all Dirichlet L-series associated with characters.

Although my program is able to compute zeros of Dirichlet L-series without outside help, I used Michael Rubinstein's L-function calculator [3] to compute an initial approximation of the zeros I wanted, which were then refined (to 20 digits after the decimal point) using the PARI/GP calculator.

The character associated to the zeros in each (compressed) file given below is fully described in the header of that file. Only zeros with positive imaginary part are given.

Tables with approximate values of the first zeros of some Dirichlet L-series, last update made on August 7, 2007

Primitive characters:

• The first 10000 zeros of the primitive character [125k, compressed with gzip]
  001-000
• The first 10000 zeros of the primitive character [125k, compressed with gzip]
  003-001
• The first 10000 zeros of the primitive character [125k, compressed with gzip]
  004-001
• The first 10000 zeros of the primitive characters [125k each, compressed with gzip]
  005-001 005-002 005-003
• The first 10000 zeros of the primitive characters [125k each, compressed with gzip]
  007-001 007-002 007-003 007-004 007-005
• The first 10000 zeros of the primitive characters [125k each, compressed with gzip]
  008-001 008-003
• The first 10000 zeros of the primitive characters [125k each, compressed with gzip]
  009-001 009-002 009-004 009-005
• The first 10000 zeros of the primitive characters [125k each, compressed with gzip]
  011-001 011-002 011-003 011-004 011-005 011-006 011-007 011-008 011-009
• The first 10000 zeros of the primitive character [125k, compressed with gzip]
  012-003
• The first 10000 zeros of the primitive characters [125k each, compressed with gzip]
  013-001 013-002 013-003 013-004 013-005 013-006 013-007 013-008 013-009 013-010 013-011
• The first 10000 zeros of the primitive characters [124k each, compressed with gzip]
  015-001 015-003 015-006
• The first 10000 zeros of the primitive characters [124k each, compressed with gzip]
  016-002 016-003 016-006 016-007
• The first 10000 zeros of the primitive characters [124k each, compressed with gzip]
  020-001 020-003 020-007
• The first 10000 zeros of the primitive characters [124k each, compressed with gzip]
  024-001 024-003
• The first 10000 zeros of the primitive characters [124k each, compressed with gzip]
  040-003 040-004 040-005 040-008 040-014 040-015
• The first 10000 zeros of the primitive characters [124k each, compressed with gzip]
  060-001 060-004 060-013
• The first 10000 zeros of the primitive characters [124k each, compressed with gzip]
  120-004 120-009 120-014 120-016 120-026 120-029

Non-primitive characters:

• The first 10 zeros of the non-primitive character [1k, compressed with gzip]
  002-000
• The first 10 zeros of the non-primitive character [1k, compressed with gzip]
  003-000
• The first 10 zeros of the non-primitive character [1k, compressed with gzip]
  004-000
• The first 10 zeros of the non-primitive character [1k, compressed with gzip]
  005-000
• The first 10 zeros of the non-primitive characters [1k each, compressed with gzip]
  006-000 006-001
• The first 10 zeros of the non-primitive character [1k, compressed with gzip]
  007-000
• The first 10 zeros of the non-primitive characters [1k each, compressed with gzip]
  008-000 008-002
• The first 10 zeros of the non-primitive characters [1k each, compressed with gzip]
  009-000 009-003
• The first 10 zeros of the non-primitive characters [1k each, compressed with gzip]
  010-000 010-001 010-002 010-003
• The first 10 zeros of the non-primitive character [1k, compressed with gzip]
  011-000
• The first 10 zeros of the non-primitive characters [1k each, compressed with gzip]
  012-000 012-001 012-002
• The first 10 zeros of the non-primitive character [1k, compressed with gzip]
  013-000
• The first 10 zeros of the non-primitive characters [1k each, compressed with gzip]
  014-000 014-001 014-002 014-003 014-004 014-005
• The first 10 zeros of the non-primitive characters [1k each, compressed with gzip]
  015-000 015-002 015-004 015-005 015-007
• The first 10 zeros of the non-primitive characters [1k each, compressed with gzip]
  016-000 016-001 016-004 016-005
• The first 10 zeros of the non-primitive characters [1k each, compressed with gzip]
  020-000 020-002 020-004 020-005 020-006
• The first 10 zeros of the non-primitive characters [1k each, compressed with gzip]
  024-000 024-002 024-004 024-005 024-006 024-007
• The first 10 zeros of the non-primitive characters [1k each, compressed with gzip]
  030-000 030-001 030-002 030-003 030-004 030-005 030-006 030-007
• The first 10 zeros of the non-primitive characters [1k each, compressed with gzip]
  040-000 040-001 040-002 040-006 040-007 040-009 040-010 040-011 040-012 040-013
• The first 10 zeros of the non-primitive characters [1k each, compressed with gzip]
  060-000 060-002 060-003 060-005 060-006 060-007 060-008 060-009 060-010 060-011 060-012 060-014 060-015
• The first 10 zeros of the non-primitive characters [1k each, compressed with gzip]
  120-000 120-001 120-002 120-003 120-005 120-006 120-007 120-008 120-010 120-011 120-012 120-013 120-015 120-017 120-018 120-019 120-020 120-021 120-022 120-023 120-024 120-025 120-027 120-028 120-030 120-031

References

Additional links

• Tables of zeros of the Riemann zeta function.
• The prime pages.

Tomás Oliveira e Silva Departamento de Electrónica, Telecomunicações e Informática Universidade de Aveiro 3810-193 AVEIRO PORTUGAL				August 29, 2010
Phone: +351-234-370379 Fax: +351-234-370545		Phone (internal): 23013 Office: DET 237
E-mail address: tos@ua.pt Home page: http://www.ieeta.pt/~tos/