#### Vol. 12, No. 1, 2018

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The mean value of symmetric square $L$-functions

### Olga Balkanova and Dmitry Frolenkov

Vol. 12 (2018), No. 1, 35–59
##### Abstract

We study the first moment of symmetric-square $L$-functions at the critical point in the weight aspect. Asymptotics with the best known error term $O\left({k}^{-1∕2}\right)$ were obtained independently by Fomenko in 2003 and by Sun in 2013. We prove that there is an extra main term of size ${k}^{-1∕2}$ in the asymptotic formula and show that the remainder term decays exponentially in $k$. The twisted first moment was evaluated asymptotically by Ng with the error bounded by $l{k}^{-1∕2+ϵ}$. We improve the error bound to ${l}^{5∕6+ϵ}{k}^{-1∕2+ϵ}$ unconditionally and to ${l}^{1∕2+ϵ}{k}^{-1∕2}$ under the Lindelöf hypothesis for quadratic Dirichlet $L$-functions.

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symmetric square $L$-functions, weight aspect, Gauss hypergeometric function, Liouville–Green method, WKB approximation