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Abstract

Asymptotic formulae for the twisted first moment of central values of symmetric square $L$-functions with harmonic weight in the weight aspect are obtained. The result extends some results in the literature. It is applied to derive an asymptotic formula for the first moment of central values of symmetric square $L$-functions without harmonic weight, under the assumption that the $L$-function is non-negative at the center of the critical strip. Analogous new formulae without harmonic weight for the first and second moments of Hecke $L$-functions are proved.