Special Values of L-functions Associated with the Space of Quadratic Forms and the Representation of Sp(2n, Fp) in the Space of Siegel Cusp Forms

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This chapter discusses special values of L-functions associated with the space of quadratic forms and the representation of Sp(2n, Fp) in the space of Siegel cusp forms. Siegel established an ingenious method of evaluating the special values at non-positive integers of partial zeta functions for totally real fields and Klingen–Siegel proved that they are rational numbers. The chapter presents special values at non-positive integers of two kinds of L-functions, one of which is the one introduced by Hashimoto, associated with the ternary zero form x1x2−x122. It also presents certain zeta functions with a kind of Gauss sums attached to the space of quadratic forms.