Distribution of values of $L'/L(\sigma,\chi_D)$

We study the distribution of values of $L'/L(\sigma,\chi_D)$, where $\sigma$ is real $>\frac12$, $D$ a fundamental discriminant, and $\chi_D$ the real character attached to $D$. In particular, assuming the GRH, we prove that for each $\sigma>1/2$ there is a density function $\mathcal Q_\sigma$ with the property that for real numbers $\alpha\leq\beta$, we have
\begin{equation*} \begin{split} \#\{D~{\text fundamental\ discriminants}\ &{\text such\ that}\ |D|\leq Y,\ {\text and}\\ &\alpha\leq\frac{L'}L(\sigma,\chi_D)\leq\beta\}\sim\frac6{\pi^2\sqrt{2\pi}}Y\int_\alpha^\beta\mathcal Q_\sigma(x)\,dx. \end{split} \end{equation*}
Our work is based on and strongly motivated by the earlier work of Ihara and Matsumoto [7].