Paper detail

Computation of the secondary zeta function

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