Geodesic
On the Distribution of Values of $L(1,f)$
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 19, Tome 302 (2003), pp. 135-148 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $S_k(N)^+$ be the set of newforms of weight $k$ for $\Gamma_0(N)$, and let $L(s,f)$, $f\in S_k(N)^+$, be the Hecke $L$-function of the form $f$. It is proved that for every integer $m\ge1$, $k=2$ and $N=p\to\infty$ $$ \sum_{f\in S_2(N)^+}\,L^m(1,f)=\frac{1}{12}B_m N+O(N^{1-\alpha}), $$ where $B_m$ is a constant defined in the paper, and $\alpha=\alpha(m)>0$ is a certain constant. This result implies the existence of the distribution function of the sequence $$ \{L(1,f),\,f\in S_2(N)^+\},\quad N=p\to\infty, $$ and also yields an explicit expression for the corresponding characteristic function. 
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     title = {On the {Distribution} of {Values} of~$L(1,f)$},
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O. M. Fomenko. On the Distribution of Values of $L(1,f)$. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 19, Tome 302 (2003), pp. 135-148. http://geodesic.mathdoc.fr/item/ZNSL_2003_302_a8/

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