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An explicit formula for the mean value of $\vert L(1,\chi)\vert^2$ is known,where $\chi$ runs over all odd primitive Dirichlet characters of primeconductors $p$. Bounds on the relative class number of the cyclotomic field${\mathbb Q}(\zeta_p)$ follow. Lately the authors obtained that the mean valueof $\vert L(1,\chi)\vert^2$ is asymptotic to $\pi^2/6$, where $\chi$ runs overall odd primitive Dirichlet characters of prime conductors $p\equiv 1\pmod{2d}$which are trivial on a subgroup $H$ of odd order $d$ of the multiplicativegroup $({\mathbb Z}/p{\mathbb Z})^*$, provided that $d\ll\frac{\log p}{\log\logp}$. Bounds on the relative class number of the subfield of degree$\frac{p-1}{2d}$ of the cyclotomic field ${\mathbb Q}(\zeta_p)$ follow. Here,for a given integer $d_0>1$ we consider the same questions for thenon-primitive odd Dirichlet characters $\chi'$ modulo $d_0p$ induced by the oddprimitive characters $\chi$ modulo $p$. We obtain new estimates for Dedekindsums and deduce that the mean value of $\vert L(1,\chi')\vert^2$ is asymptoticto $\frac{\pi^2}{6}\prod_{q\mid d_0}\left (1-\frac{1}{q^2}\right )$, where$\chi$ runs over all odd primitive Dirichlet characters of prime conductors $p$which are trivial on a subgroup $H$ of odd order $d\ll\frac{\log p}{\log\logp}$. As a consequence we improve the previous bounds on the relative classnumber of the subfield of degree $\frac{p-1}{2d}$ of the cyclotomic field${\mathbb Q}(\zeta_p)$. Moreover, we give a method to obtain explicit formulasand use Mersenne primes to show that our restriction on $d$ is essentiallysharp.

Mean square values of $L$-functions over subgroups for non primitive characters, Dedekind sums and bounds on relative class numbers