## Rankin triple $$L$$-functions.(English)Zbl 0637.10023

Let $$k$$ be an algebraic number field and $$K$$ be either a cubic extension of $$k$$ or the product of $$k$$ and a quadratic extension of $$k$$. Let $$G=\text{PGL}(2,K)$$. The $$L$$-group of $$G$$ has an obvious 8-dimensional representation $$r$$. Corresponding to an irreducible cuspidal representation $$\pi$$ of $$G(\mathbb A)$$ there is Langland’s $$L$$-function $$L(\pi,r)$$.
In this paper the authors give an integral representation of $$L(\pi,r)$$ involving an Eisenstein series on $$\text{PSp}(6)$$ (observe that $$G$$ has a finite covering which can be imbedded into $$\text{PSp}(6)$$ ). From a detailed study of this integral and the properties of the Eisenstein series they deduce the analytic continuation of $$L(\pi,r)$$ with exact location of the poles.

### 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 11R39 Langlands-Weil conjectures, nonabelian class field theory 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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### References:

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