Assuming the RH and $\rho_n =\frac12 + \gamma_n i$ being the n-th non-trivial zero of $\zeta(s)$, then numerical evidence suggests that:
$$f(s) :=\displaystyle \sum_{n=1}^{\infty} \left(\frac{1}{\rho_n^s} +\frac{1}{\overline{\rho_n}^s}\right)$$
converges for all $s \ge 1$ and has an infinite number of zeros 'just above' the odd integer values of $s$.
I know that $f(1) = 1 + \frac{\gamma}{2} -\frac12 \ln(4\,\pi)$, however wondered whether there is anything known about:
1) closed forms for other positive integer values of $s$ and/or,
2) the (quite regular) pattern of the zeros of $\,f(s)$?
I attach a graph to illustrate the point around the zeros ($y=f(s)$, magnified by $10^{12}$ and $n=99$):
