CFKRS–DGH conjecture for the first moment of quadratic Dirichlet L-functions

Let $X$ be large, let $\Phi$ be the smooth test function used in the moment, and let $\alpha$ be complex. Define

$$\widetilde{\Phi}(s)=\int_0^\infty \Phi(x)x^{s-1}\,dx$$

and let $\zeta_2(s)=(1-2^{-s})\zeta(s)$. Let $\gamma_\alpha$ be the factor defined by the approximate functional equation, and define $B_\alpha$ by

$$\zeta_2(1+2\alpha)B_\alpha:=\sum_{(n,2)=1}\frac{1}{n^{1+2\alpha}}\prod_{p\mid n}(1+p^{-1})^{-1}.$$

CFKRS–DGH conjecture. The first moment satisfies

$$\sideset{}{^*}\sum_{(d,2)=1}L\left(\tfrac12+\alpha,\chi_{8d}\right)\Phi\left(\frac dX\right)=\frac{X\widetilde{\Phi}(1)}{2\zeta_2(2)}\zeta_2(1+2\alpha)B_\alpha+\frac{X^{1-\alpha}\widetilde{\Phi}(1-\alpha)\gamma_\alpha}{2\zeta_2(2)}\zeta_2(1-2\alpha)B_{-\alpha}+O\left(X^{1/2+\varepsilon}\right).$$

Here $B_\alpha$ has an absolutely convergent Euler product for $\alpha$ in a neighborhood of the origin, and the estimate is uniform for $|\operatorname{Re}(\alpha)|\ll(\log X)^{-1}$ and $|\operatorname{Im}(\alpha)|\ll(\log X)^2$. This is the conjectural asymptotic underlying the first-moment calculation and is attributed in the source to CFKRS and DGH; its resolution is not specified in the paper.