A computable formula for evaluating the mean square sum of $L$-functions

Abstract

For Dirichlet characters $\chi$ mod $k$ where $k\ge 3$, we here give a computable formula for evaluating the mean square sums $\sum _{\begin{array}{c}\chi \text{ mod }k\\ \chi (-1)=(-1{)}^r\end{array}}|L(r,\chi ){|}^2$ for any positive integer $r\ge 3$. We also give an inductive formula for computing the sum $\sum _{\begin{array}{c}1\le m\le k\\ (m,k)=1\end{array}}\frac{1}{{\left(\sin \left(\frac{\pi m}{k}\right)\right)}^{2n}}$ where $n$ is a positive integer in terms of Bernoulli numbers and binomial coefficients.