Below is a partially scaled plot of $\vartheta (\gamma_n) - \pi (n - 3/2) ,$ where $\gamma_n$ is the imaginary part of the $n$th zeta zero, and $\vartheta $ is the Riemann-Siegel theta function, for the first $2$m zeros:
It certainly looks to have standard normal distribution, is this indeed the case?
Note
Cross-posted and answered here