The mean value of symmetric square L-functions

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(k(-1/2)) 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 lk(1/2+epsilon). We improve the error bound to l(5/6+epsilon)k(-1/2+epsilon) unconditionally and to l(-1/2+epsilon)k(-1/2) under the Lindelof hypothesis for quadratic Dirichlet L-functions.