Geodesic
On the behavior of automorphic $L$-functions at the center of the critical strip
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 17, Tome 276 (2001), pp. 300-311 
O. M. Fomenko. On the behavior of automorphic $L$-functions at the center of the critical strip. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 17, Tome 276 (2001), pp. 300-311. http://geodesic.mathdoc.fr/item/ZNSL_2001_276_a14/
@article{ZNSL_2001_276_a14,
     author = {O. M. Fomenko},
     title = {On the behavior of automorphic $L$-functions at the center of the critical strip},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {300--311},
     year = {2001},
     volume = {276},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2001_276_a14/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Let $\mathscr F$ be the Hecke eigenbasis of the space $S_2(\Gamma_0(p))$ of $\Gamma_0(p)$-cusp forms of weight 2. Let $p$ be a prime. Let $\mathscr H_f(s)$ be the Hecke $L$-series of form $f\in\mathscr F$. The following statements are proved: $$ \sum_{f\in\mathscr F}\mathscr H_f\left(\frac12\right)=\zeta(2)\frac p{12}+O\left(p^{\frac{31}{32}+\varepsilon}\right) $$ and $$ \sum_{f\in F}\mathscr H_f\left(\frac12\right)^2=\frac{\zeta(2)^3}{\zeta(4)}\frac p{12}\log p+O(p\log\log p). $$ We also give a correct proof of a previous author's theorem on automorphic $L$-functions.