Keating–Snaith conjecture on logarithmic central values

Let $\mathcal{F}$ be a family of $L$-functions of type of symmetry $\mathrm{SO(even)}$ or $\mathrm{Sp}$, and let $\mathcal{F}_X$ be a finite truncation increasing to $\mathcal{F}$ as $X$ grows. Keating–Snaith conjecture. There exist $M_{\mathcal{F}_X}$ and $V_{\mathcal{F}_X}$ such that, for any positive real numbers $\alpha<\beta$,

$$\frac{1}{|\mathcal{F}_X|}\left|\left\{f\in\mathcal{F}_X:\frac{\log L(\frac12)-M_{\mathcal{F}_X}}{V_{\mathcal{F}_X}}\in(\alpha,\beta)\right\}\right|\xrightarrow[X\to\infty]{}\frac{1}{\sqrt{2\pi}}\int_\alpha^\beta e^{-x^2/2}\,dx.$$

Thus the logarithms of the central values asymptotically follow a normal distribution with mean $M_{\mathcal{F}_X}$ and variance $V_{\mathcal{F}_X}$.