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 prime conductors p. Bounds on the relative class number of the cyclotomic field
${\mathbb Q}(\zeta _p)$
follow. Lately, the authors obtained that the mean value of
$\vert L(1,\chi )\vert ^2$
is asymptotic to
$\pi ^2/6$
, where
$\chi $
runs over all 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 multiplicative group
$({\mathbb Z}/p{\mathbb Z})^*$
, provided that
$d\ll \frac {\log p}{\log \log p}$
. 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 the nonprimitive odd Dirichlet characters
$\chi '$
modulo
$d_0p$
induced by the odd primitive characters
$\chi $
modulo p. We obtain new estimates for Dedekind sums and deduce that the mean value of
$\vert L(1,\chi ')\vert ^2$
is asymptotic to
$\frac {\pi ^2}{6}\prod _{q\mid d_0}\left (1-\frac {1}{q^2}\right )$
, where
$\chi $
runs over all odd primitive Dirichlet characters of prime conductors p which are trivial on a subgroup H of odd order
$d\ll \frac {\log p}{\log \log p}$
. As a consequence, we improve the previous bounds on the relative class number 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 formulas and use Mersenne primes to show that our restriction on d is essentially sharp.