  Assuming the Generalized Riemann Hypothesis, we establish explicit bounds in
the $q$-aspect for the logarithmic derivative
$\left(L'/L\right)\left(\sigma,\chi\right)$ of Dirichlet $L$-functions, where
$\chi$ is a primitive character modulo $q\geq 10^{30}$ and
$1/2+1/\log{\log{q}}\leq\sigma\leq 1-1/\log\log q$. In addition, for $\sigma=1$
we improve upon the result by Ihara, Murty and Shimura (2009). Similar results
for the logarithmic derivative of the Riemann zeta-function are given.
