In his proof of the prime number theorem Littlewood claims that the sum $\sum_{\rho} |\Gamma(\rho)|$ converges, where ${\rho}$ ranges over the non-trivial zeros of $\zeta$ counting multiplicity.
I guess this can be proven using Stirling's formula and a result on the growth of the modulus of zeros of $\zeta$.
Most of the results in this direction that I have seen are claims on $n(T)= \sharp(\zeta^{-1}(\{ 0\})\cap \{ z\in \mathbb C,\, |z-1/2| < T \})$, with $T>0$. For instance, from Jensen's formula, one has $n(T) \leq 3T \log T$ for $T\gg 0$. I do not see how can one derive a proof of the convergence of the aforementioned sum using such bounds.
Does such proof exist or do you know any result of the form $|\rho_k|\geq f(k)$ where $(\rho_k)_{k\in \mathbb N}$ are the non-trivial zeros of $\zeta$ ordered by modulus counting multiplicty which is sufficient to prove the convergence ?