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. Let f be a Maass form for SL $$(3, {\mathbb{Z}})$$ which is fixed and uj be an orthonormal basis of even Maass forms for SL $$(2, {\mathbb{Z}})$$ , we prove an asymptotic formula for the average of the product of the Rankin–Selberg L-function of f and uj and the L-function of uj at the central value 1/2. This implies simultaneous nonvanishing results of these L-functions at 1/2.
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