On the gamma factor of the triple \(L\)-function. I
Let \(\pi_1,\pi_2,\pi_3\) be the components at some archimedean place of three irreducible cuspidal automorphic representations of \(\text{GL}(2)\) and let \(\sigma_i\) be the 2-dimensional complex representation of the Weil group \(W_{\mathbb R}\) or \(W_{\mathbb {C}}\) corresponding to \(\pi_i\). It is proved in the paper that the local \(L\)-factor attached to \((\pi_1,\pi_2,\pi_3)\), which was defined as a greatest common divisor of local integrals, is equal to \(L(s,\sigma_1\otimes\sigma_2\otimes\sigma_3)\). This can be derived from the local functional equation, except when \(\pi_1,\pi_2,\pi_3\) are all class-one principal series representations. In that case an explicit computation of the local integral gives the result. See also Part II [J. Reine Angew. Math. 499, 199-223 (1998; Zbl 0923.11078)].
irreducible cuspidal automorphic representations of GL(2)
scientific article; zbMATH DE number 1425206
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