An explicit formula for the quadratic mean value at s = 1 of the Dirichlet L-functions associated with the odd Dirichlet characters modulo f > 2 is known. Here we present a situation where we could prove an explicit formula for the quadratic mean value at s = 1 of the Dirichlet L-functions associated with the odd Dirichlet characters modulo not necessarily prime moduli f > 2 that are trivial on a subgroup H of the multiplicative group (Z/f Z)∗. This explicit formula involves summation S(H, f ) of Dedekind sums s(h, f ) over the h ∈ H. A result on some cancelation of the denominators of the s(h, f )’s when computing S(H, f ) is known. Here, we prove that for some explicit families of f ’s and H’s this known result on cancelation of denominators is the best result one can expect. Finally, we surprisingly prove that for p a prime, m ≥ 2 and 1 ≤ n ≤ m/2, the values of the Dedekind sums s(h, pm) do not depend on has h runs over the elements of order pn of the multiplicative cyclic group (Z/pmZ)∗.