The Distribution of Values of $\frac{L^{\mathrm{\prime }}}{L}(1\mathrm{/}2+ϵ,{\chi }_D)$

We determine the limiting distribution of the family of values $\frac{L^{\mathrm{\prime }}}{L}(1\mathrm{/}2+ϵ,{\chi }_D)$ as $D$ varies over fundamental discriminants. Here, $0<ϵ<\frac{1}{2}$, and ${\chi }_D$ is the real character associated with $D$. Moreover, we also establish an upper bound for the rate of convergence of this family to its limiting distribution. As a consequence of this result, we derive an asymptotic bound for the small values of $\mid \frac{L^{\mathrm{\prime }}}{L}(1\mathrm{/}2+ϵ,{\chi }_D)\mid$.