  Finding the mean square averages of the Dirichlet $L$-functions over
Dirichlet characters $\chi$ of same parity is an active problem in number
theory. Here we explicitly evaluate such averages of $L(3,\chi)$ and
$L(4,\chi)$ using certain trigonometric sums and Bernoulli polynomials and
express them in terms of the Euler totient function $\phi$ and the Jordan
totient function $J_s$.
