On triple product L-functions

Let $π=π_1 \otimes π_2 \otimes π_3$ be a unitary cuspidal automorphic representation of $\mathrm{GL}_3^3(\mathbb{A}_F)$ where $F$ is a number field. Assume that $π$ is everywhere tempered. Under suitable local hypotheses, for a sufficiently large finite set of places $S$ of $F$ we prove that the triple product $L$-function $L^S(s,π,\otimes^3)$ admits a meromorphic continuation to $\mathrm{Re}(s) >\tfrac{1}{2}$. We also give some information about the possible poles.

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