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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Values of $ L$-functions at the critical point


Author: Liem Mai
Journal: Proc. Amer. Math. Soc. 122 (1994), 415-428
MSC: Primary 11F67; Secondary 11M41
MathSciNet review: 1227525
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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:

$\displaystyle {\varepsilon _D}(n) = \sum\limits_{\begin{array}{*{20}{c}} {g > 0... ...^2}) = 1} \\ \end{array} } {\left( {\frac{{{D_0}}}{{{g^{ - 2}}n}}} \right)} \,g$

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)$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1227525-0
PII: S 0002-9939(1994)1227525-0
Keywords: L-functions, discriminants
Article copyright: © Copyright 1994 American Mathematical Society