On the gamma factor of the triple L-function, I

Abstract

Introduction. The study of the triple L-function began with Garrett [2]. From a representation-theoretic point of view, Piatetski-Shapiro and Rallis [9] and the author [6] defined the archimedean or nonarchimedean L-factor as the greatest common divisor (GCD) of local integrals. The purpose of this paper is to prove that the archimedean GCD L-factor agrees with the expected L-factor up to invertible functions if the local representation comes from a cuspidal automorphic representation. This theorem follows from the local functional equation by formal argument except when the representation is unramified. To treat the unramified case, we carry out the explicit calculation of the unramified local integral. Stimulated by Stade [12], we make use of a hypergeometric function 3F2 and its two-term relations and three-term relation. We also need a theorem on some infinite integral involving Bessel functions, proved by Bailey [1] in 1936. The key lemma is the following (see Lemma 2.4).