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

Abstract

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