  We consider the logarithm of the central value $\log L(1/2)$ in the
orthogonal family ${L(s,f)}_{f \in H_k}$ where $H_k$ is the set of weight $k$
Hecke-eigen cusp form for $SL_2(\mathbb{Z})$, and in the symplectic family
${L(s,\chi_{8d})}_{d \asymp D}$ where $\chi_{8d}$ is the real character
associated to fundamental discriminant $8d$. Unconditionally, we prove that the
two distributions are asymptotically bounded above by Gaussian distributions,
in the first case of mean $-\frac{1}{2} \log \log k$ and variance $\log \log
k$, and in the second case of mean $\frac{1}{2}\log \log D$ and variance $\log
\log D$. Assuming both the Riemann and Zero Density Hypotheses in these
families we obtain the full normal law in both families, confirming a
conjecture of Keating and Snaith.
