# Derivative of Riemann Zeta at nontrivial zeros

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.

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