  In this paper we consider two spectral realizations of the zeros of the
Riemann zeta function. The first one involves all non-trivial (non-real) zeros
and is expressed in terms of a Laplacian intimately related to the prolate wave
operator. The second spectral realization affects only the critical zeros and
it is cast in terms of sheaf cohomology. The novelty is that the base space is
the Scaling Site playing the role of the parameter space for the $\zeta$-cycles
and encoding their stability by coverings.
