A sum rule for the critical zeros of $\zeta(s)$

Abstract

The two-parameter series over the critical zeros of the Riemann Zeta function $Re\sum_{\rho}\frac{x^{(\rho-a)/4a}}{\sqrt{\rho-a}\sinh[\frac{\pi}{2}\sqrt{\frac{\rho-a}{a}}]\zeta'(\rho)}$ is evaluated in terms of $\zeta(s)$ on the real axis.