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

Abstract

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