A survey on the evaluation of the values of Dirichlet $L$-functions and of their logarithmic derivatives on the line of $1$

Let $q$ be a positive integer $q>1$, and let $\chi$ be a Dirichlet character modulo $q$. Let $L(s, \chi)$ be the attached Dirichlet $L$-functions,
and let $L^\prime(s, \chi)$ denote its derivative with respect to the complex variable $s$. In this talk, we survey certain known results on the evaluation of  values of  Dirichlet $L$-functions  and  of their logarithmic derivatives at $1+it_0$ for fixed real number $t_0$.
We also give a new asymptotic formula for the $2k$-th power mean value of $\left|(L^\prime/L)(1+it_0, \chi)\right|$ when $\chi$ runs over all Dirichlet characters modulo $q>1$, for any fixed real number $t_0$. This is joint work with professor Kohji Matsumoto.