The author considers the analytic properties of the L-function \(L(s,\pi,r^ 0_ 3)\) where \(\pi\) is a cuspidal automorphic representation of \(GL_ 2(F_{{\mathbb{A}}})\) (F a number field) and \(r^ 0_ 3=r_ 3\otimes (\Lambda^ 2r_ 3)^{-1}\) where generally \(r_ j\) is the j-th symmetric power of the standard representation of \(GL_ 2({\mathbb{C}})\). This arises in the global theory of automorphic forms in two different contexts.
First of all it is L(s,\(\pi\) \(\times \Pi)/L(s,\pi)\) where L(s,\(\pi\) \(\times \Pi)\) is the Rankin-Selberg convolution on \(GL_ 3(F_{{\mathbb{A}}})\) of Jacquet-Shalika and L(s,\(\pi)\) is the Hecke L- function. Moreover \(\Pi\) is the Gelbart-Jacquet adjoint square lift of \(\pi\) to \(GL_ 3(F_{{\mathbb{A}}})\). On the other hand the gamma-function corresponding to \(L(s,\pi,r^ 0_ 3)\) appears in the local coefficients of Whittaker functions for representations of a split group of type \(G_ 2\) and hence appears in the author’s theory of such functions. Combining these he is able to prove a functional equation for \(L(s,\pi,r^ 0_ 3)\) and \(L(s,\pi,r_ 3)\) and he gives a very straightforward criterion for checking that these functions are entire. He shows that this criterion applies to a large class of cuspidal holomorphic modular cusp forms of weight 2.
The argument also leads to an explicit computation of the Plancherel constant for representations of a split group of type \(G_ 2\) induced from \(GL_ 2\) over a local field of characteristic 0. He also determines the nature of the Jordan-Hölder series for such representations.
The paper concludes with a nonvanishing result for a partial L-function \(L_ S(s,\pi,r_ 5)\) and some applications. This uses a similar method but with a group of type \(F_ 4\) in place of \(G_ 2\).