  In this paper we study the spectrum of a fundamental differential operator on
a Hilbert-P\'olya space. A number is an eigenvalue of this differential
operator if and only if it is a nontrivial zero of the Riemann zeta function.
An explicit formula is given for the eigenfunction associated with each
nontrivial zero of the zeta function. Every eigenfunction is characterized via
the Poisson summation formula by a sequence of mysterious functions whose
explicit formulas are given.
