  The zeta-function of a manifold is closely related to, and sometimes can be
calculated completely, in terms of its periods. We report here on a practical
and computationally rapid implementation of this procedure for families of
Calabi-Yau manifolds with one complex structure parameter phi. Although partly
conjectural, it turns out to be possible to compute the matrix of the Frobenius
map on the third cohomology group of X(phi) directly from the Picard-Fuchs
differential operator of the family. To illustrate our method, we compute
tables of the quartic numerators of the zeta-functions for six manifolds of
increasing complexity as the parameter phi varies in Fp. For four of these
manifolds, we do this for the 500 primes p=5,7,...,3583, while for two
manifolds we extend the calculation to 1000 primes. The tables for 5 <= p <= 97
are part of this article while the remaining tables are attached in electronic
form. Interest attaches to the cases for which the numerators factorise. Some
of these factorisations can be associated with parameter values for which the
underlying manifold becomes singular. For the cases we consider here, the
singularities are all of conifold or hyperconifold type. In these cases the
numerator degenerates to a cubic and this factorises into the product of a
linear and a quadratic factor. The quadratic term contains a coefficient that
is the p'th coefficient of a modular form. Some of our examples have
singularities when the parameter satisfies a polynomial equation that does not
factorise over Q. When this happens, the corresponding forms are modular forms
with neben type or Hilbert modular forms. The numerator can also factorise into
two quadrics. This happens when the Hodge structure of the manifold splits,
sometimes this happens for algebraic values of the parameter and we identify,
in this way, attractor points of rank two of the parameter space.
