Q-series identities and values of certain L-functions

As usual, define Dedekind's eta-function η(z) by the infinite product η(z}:=q^1/24 ∏_n=1^∞ (1-qⁿ) (q:=e^2πix throughout). In a recent paper, D. Zagier proved that (note: empty products equal 1 throughout) ∑_n=0^∞(η(24x)-q(1-q²⁴)(1-q ⁴⁸)⋯(1-q²⁴ⁿ))=η(24x))D(q)+E(q), where the series D(q) and E(q) are defined by D(q) = -1/2+∑_n=1^∞ q²⁴ⁿ/1-q²⁴ⁿ = -1/2+∑_n=1^∞d(n)q²⁴ⁿ = -1/2+q²⁴+2q⁴⁸+2q⁷²+3q⁹⁶+... E(q) = 1/2∑_n=1^∞(12/n)nqⁿ²=1/2q-5/2q ²⁵-7/2q⁴⁹+11/2q¹²¹+... 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 χ₂ is the Kronecker character for Q(√2), these identities can be used to show that -2e^-t/8 ∑_n=0^∞ (1-e^-2t) (1-e^-4t)⋯(1-e^-2nt)/1+e^-t(1+e ^-3t)⋯(1+e^-(2n+1)t) = ∑_n=0^∞(-1/8)ⁿ ·L(χ₂,-2n-1)·tⁿ/n