@ Tom Copeland In fact my numbers coincide with those of Helm, as you can check from my values.

@ Tom Copeland No, my numbers is those in which you are interested. If $\rho = \frac12+i\gamma$ is a zero of zeta then I compute $\sum_{n=1}^\infty \gamma_n^{-s}$.

@Tom Copeland Sage is free. from mpmath import *, load mpmath to Sage. mp.dps=50, put your computation to 50 digits. secondzeta(s) compute the value of this function at the point s. Even in cases the series is divergent, because it computes the only meromorphic extension.

There is a book by Srivastava, Hari M., Junesang Choi, "Series Associated with the Zeta and Related Functions" that contains many series. Springer 2001. But is is at the Faculty at this time of coronavirus and it is only what I have in mind,

In your example a good "envelope" appear to be $4(1+x^2)/Q $ with $Q=2+2x^2+x^4-\sqrt{(x^2-1)^2+16x^2}$.

@pisoir Thanks for the edits

@domotorp In the paper we show also that the sums are not bounded for any irrational.

@theonetruepath It is in Elsevier but I can download it without any problem. The second paper that of Tutte has a pay wall, yes.

Section 14.25 of Titchmarsh, treats the case assuming RH. But do not get what you want.

@jjcale Yes you have reason, I was wrong.