The convergence of this particular type of sums over non-trivial zeros of the zeta function has been investigated: $$\sum_{\rho}\frac{1}{\rho}=1+\frac{1}{2}\gamma -\frac{1}{2}\ln 4\pi ,$$ $$\sum_{\rho}\frac{1}{\rho ^2}=1+\gamma ^2-\frac{1}{8}\pi ^2+2\gamma _1 ,$$ $$\sum_{\rho}\frac{1}{\rho ^3}=1+\gamma ^3+3\gamma\gamma _1 +\frac{3}{2}\gamma _2 -\frac{7}{8}\zeta (3),$$ ($\gamma _n$ are Stieltjes constants) and so on.
Has the convergence of another "type" of sums over the non-trivial zeros been investigated, or are the $\rho ^{-k}$ sums the only ones? For example, using the existing knowledge about these sums, could the convergence of $$\sum_{\rho}e^{\rho x}$$ be determined?
Maybe you could refer me to some books or another sources (if some exist) where there is a broader discussion of convergence of such sums apart from the $\rho ^{-k}$ ones (and Riemann's explicit formula).