Abstract :
Using the Jordan algebras methods, specially the properties of Peirce decomposition and the
Frobenius transformation, we compute the coefficients of the zeta functional equation, in the case
of Jordan algebras of type II. As particular cases of our result, we can cite the case of V = M(n,R)
studied by Gelbart [Mem. Amer. Math. Soc. 108 (1971)] and Godement and Jacquet [Zeta functions
of simple algebras, Lecture Notes in Math., vol. 260, Springer-Verlag, Berlin, 1972], and the case of
V = Herm(3,Os ) studied by Muro [Adv. Stud. Pure Math. 15 (1989) 429]. Let us also mention, that
recently, Bopp and Rubenthaler have obtained a more general result on the zeta functional equation
by using methods based on the algebraic properties of regular graded algebras which are in oneto-
one correspondence with simple Jordan algebras [Local Zeta Functions Attached to the Minimal
Spherical Series for a Class of Symmetric Spaces, IRMA, Strasbourg, 2003]. The method used in
this paper is a direct application of specific properties of Jordan algebras of type II.
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