  We define, answering a question of Sarnak in his letter to Bombieri, a
symplectic pairing on the spectral interpretation (due to Connes and Meyer) of
the zeroes of Riemann's zeta function. This pairing gives a purely spectral
formulation of the proof of the functional equation due to Tate, Weil and
Iwasawa, which, in the case of a curve over a finite field, corresponds to the
usual geometric proof by the use of the Frobenius-equivariant Poincar\'e
duality pairing in etale cohomology. We give another example of a similar
construction in the case of the spectral interpretation of the zeroes of a
cuspidal automorphic $L$-function, but this time of an orthogonal nature. These
constructions are in adequation with Deninger's conjectural program and the
arithmetic theory of random matrices.
