Values of $L$-functions at the critical point

Abstract:

For a discriminant D of a binary quadratic form, we study the average value of $L(s,{\varepsilon _D})$ at the critical point $\frac {1}{2}$ where ${\varepsilon _D}$ is defined by W. Kohnen and D. Zagier: \[ {\varepsilon _D}(n) = \sum \limits _{\begin {array}{*{20}{c}} {g > 0} \\ {g|\delta ,{g^2}|n} \\ {(\delta /g,n/{g^2}) = 1} \\ \end {array} } {\left ( {\frac {{{D_0}}}{{{g^{ - 2}}n}}} \right )} g\] for $n \in \mathbb {N}$ and $D = {D_0}{\delta ^2},{D_0}$ a fundamental discriminant and $\delta \in \mathbb {N}$. When $D = {D_0},L(s,{\varepsilon _{{D_0}}})$ is the Dirichlet series $L(s,(\frac {{{D_0}}}{ \cdot }))$. We derive an asymptotic formula for $\sum \nolimits _D {L(\frac {1}{2},{\varepsilon _D})}$, where the sum runs over all discriminants $D \in (0,Y]$ or $[ - Y,0)$.