As usual, let $L(s,\chi) = \sum_{n\geq 1} \chi(n)n^{-s}$ be the L-function of a Dirichlet character mod $q$ and assume GRH. Since no nonzero meromorphic function has infinitely many zeroes in a compact set and overall has countably many zeroes (a consequence of the Identity Theorem), we may enumerate the zeroes in the closed upper-half plane as $\rho_n = \frac{1}{2} + i\gamma_n$, $0 \leq \gamma_0 \leq \gamma_1, \ldots$ where $\gamma_n \to \infty$ as $n\to \infty$.
Is it true that $\lim_{n\to \infty} (\gamma_{n+1} - \gamma_n) > 0$ (possibly infinite)? Is it true that if $b_n = (\log \gamma_n)/\gamma_n$, then $\lim_{n\to\infty} \frac{b_{n+1}}{b_n} < 1$? I need these to be able to show convergence of other series involving zeroes with the help of D'Alambert's ratio test (I will not give them here, as I do not think it will be helpful).
I would not be surprised if the reasonings involve facts about general meromorphic functions so do have in mind that particular properties of $L$-functions might not be needed (though feel free to use as many as you want).
Any help appreciated!