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The L-function \(L_ 3(s,\pi _{\Delta})\) is entire. (English) Zbl 0558.10025

Let \(\tau\) be the Ramanujan function and represent \(\tau\) (p) for a prime p as \(\alpha_ p+{\bar \alpha}_ p\) where \(\alpha_ p{\bar \alpha}_ p=p^{11}\). Denote by \(L_ m\) the Euler product \(\prod_{p}\prod^{m}_{j=0}(1-\alpha^ j_ p {\bar \alpha}_ p^{m-j} p^{-s})^{-1}\). The analytic properties of \(L_ 1\) (Mordell, Hecke) and \(L_ 2\) (Rankin, Selberg, Shimura) have been known for some time. These results have been of considerable significance, particularly in regard to the Ramanujan conjecture. It is generally believed that all the \(L_ m\) should have ’natural’ analytic properties, and this would be of very great interest, and would, for example, lead to a proof of the Sato-Tate conjecture.
In this paper the authors derive in an ingenious way the properties of \(L_ 3\) which represents a significant advance. The proof is based on two ingredients, the theory of Eisenstein series for a group of type \(G_ 2\) (this is based on an idea of R. P. Langlands) and then the use of the Gelbart-Jacquet lifting from \(GL_ 2\) to \(GL_ 3\) combined with the theory of L-functions for representations of \(GL_ 3\).
Reviewer: S.J.Patterson

MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11S40 Zeta functions and \(L\)-functions
11F11 Holomorphic modular forms of integral weight
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References:

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