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I would like to know whether the real part of the first derivative of the Zeta function at the non trivial zeros of Zeta is stricly positive and if so, is there a proof for it.

Also, are there tables or studies of first derivative values at zeros of the Zeta function?

Thank you all for your help.

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No, it is not.

Using sage, format is n-th zero, zero, Re(zeta'(s)):

127 (0.5 + 282.46511476505209623j) -0.051786600288820422906
136 (0.5 + 295.57325487895829239j) -0.031941907138887597722
196 (0.5 + 391.45608356363804577j) -0.047584869480501458447
213 (0.5 + 415.45521499629459886j) -0.57814342438306688421
233 (0.5 + 446.86062269642952253j) -0.27263428500279642628
256 (0.5 + 478.94218153463482654j) -0.12771720375134527005
289 (0.5 + 527.90364160127234523j) -0.96196701578032407318
368 (0.5 + 637.39719315983730717j) -0.31309354934078881436

Code:

import mpmath
for n in [ 1 .. 1000]:
    z=mpmath.zetazero(n)
    d=mpmath.zeta(z,derivative=1).real
    if d<mpmath.mpf(0):  print n,z,d

Added

As Joël reminds, sage is free software.

You can even use it in a browser on: https://cloud.sagemath.com/

As an optional package it contains a database of zeta zeros at http://sagemath.org/packages/optional/ Check: database_odlyzko_zeta.spkg

This is OEIS: https://oeis.org/A153815

A153815 Numbers of nontrivial zeros of Rieman Zeta function where the real part of Zeta'(s) becomes negative.

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  • 3
    $\begingroup$ And let me add, in case the OP wouldn't know it, that SAGE is a freely available math software that can be found on www.sagemath.org $\endgroup$ – Joël Dec 3 '13 at 15:01
  • $\begingroup$ And I made the database_odlyzko_zeta package (which is pre-installed into cloud.sagemath.com) using files from dtc.umn.edu/~odlyzko/zeta_tables $\endgroup$ – William Stein Dec 7 '13 at 14:41

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