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— In 1972, Godement and Jacquet obtained thé functional équation of thé zêta function of simple algebras. In thé case where thé groundfîeld is C, such zêta functions are associated to each K x K-fmite coefficient of an irreducible admissible représentation n of GL (n, C). In this paper we consider thé case of a family of complex symmetric spaces G/H which are realized as Zariski open subsets of a vector space (in GodementJacquets case, thé space GL(/z, C)wGL(n, C)xGL(/î, C)/GL(n, C) is embedded in thé full matrix space M (n, C)). This family of symmetric spaces corresponds to thé prehomogeneous vector spaces of commutative parabolic type. If n is a generic représentation of thé principal spherical séries of G which has a natural H-invariant generalized vector, we defîne ^ and H-invariant coefficients of K and thé zêta function associated to thèse coefficients. We obtain an explicit functional équation for this zêta function.

Fonction zêta associée à la série principale sphérique de certains espaces symétriques