Computation of the secondary zeta function

Abstract

The secondary zeta function $Z(s)=\sum_{n=1}^\infty\alpha_n^{-s}$, where $\rho_n=\frac12+i\alpha_n$ are the zeros of zeta with $\Im(\rho)>0$, extends to a meromorphic function on the hole complex plane. If we assume the Riemann hypothesis the numbers $\alpha_n=\gamma_n$, but we do not assume the RH. We give an algorithm to compute the analytic prolongation of the Dirichlet series $Z(s)=\sum_{n=1}^\infty \alpha_n^{-s}$, for all values of $s$ and to a given precision.