q-series identities and values of certain L-functions

As usual, define Dedekind's eta-function eta (z) by the infinite product [GRAPHICS] In a recent paper D. Zagier proved that (note: empty products equal 1 throughout) [GRAPHICS] where the series D(q) and E(q) are defined by [GRAPHICS] [GRAPHICS] = 1 1/2 + q(24) + 2q(48) + 2q(72) + 3q(96) +..., [GRAPHICS] Here d(n) denotes the number of positive divisors of n. We obtain two infinite families of such identities and describe some consequences for L-functions and partitions. For example, if chi (2) is the Kronecker character for Q(root2), these identities can be used to show that [GRAPHICS] [GRAPHICS]