  Let M be a compact closed n-dimensional manifold. Given a Riemannian metric
on M, we consider the zeta function Z(s) for the de Rham Laplacian and the
Bochner Laplacian on p-forms. The hessian of Z(s) with respect to variations of
the metric is given by a pseudodifferential operator T(s). When the real part
of s is less than n/2-1, we compute the principal symbol of T(s). This can be
used to determine whether the general critical metric for Z(s) or one of its s
derivatives has finite index, or whether it is an essential saddle point.
