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A mean value of a triple product of $L$-functions

Luo has proven an optimal upper bound for the $L^4$-norm of dihedral Maass forms of large eigenvalue, by bounding a mean value of triple product $L$-functions. Motivated by this result, we study a mean value of $L$-functions having similar shape, and obtain for it an asymptotic with power savings. Our work may be helpful in eventually obtaining an asymptotic for the $L^4$-norm.