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As shown by Michel-Ramakrishan (2007) and later generalized byFeigon-Whitehouse (2008), there are "stable" formulas for the average centralL-value of the Rankin-Selberg convolutions of some holomorphic forms of fixedeven weight and large level against a fixed imaginary quadratic theta series.We obtain exact finite formulas for twisted first moments of Rankin-SelbergL-values in much greater generality and prove analogous "stable" formulas whenone considers either arbitrary modular twists of large prime power level orreal dihedral twists of odd type associated to a Hecke character of mixedsignature.

Stable averages of central values of Rankin-Selberg L-functions: some new variants