  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.
