Extreme values of the Dirichlet $L$-functions at the critical points of the Riemann zeta function

We estimate large and small values of $\mathrm{\mid }L({\rho }^{\mathrm{\prime }},\chi )\mathrm{\mid }$, where $\chi$ is a primitive character mod $q$ for $q>2$ and ${\rho }^{\mathrm{\prime }}$ runs over critical points of the Riemann zeta function in the right half of the one-line, that is, the points where ${\zeta }^{\mathrm{\prime }}({\rho }^{\mathrm{\prime }})=0$ and $1\le \mathrm{\Re }{\rho }^{\mathrm{\prime }}$. It would be interesting to study how a certain Dirichlet $L$-function behaves at the critical points of the Riemann zeta function. We expect extreme values that an $L$-function would take at the critical points of the Riemann zeta function to be very close to the extreme values that the $L$-function would otherwise take to the right of the vertical line $\mathrm{\Re }s=1$. That is, an $L$-function is expected to behave in a manner that is independent of the nature of the points that are special with respect to the Riemann zeta function. The results obtained in this paper corroborate this behavior.