  Let M be a compact manifold without boundary. Associated to a metric g on M
there are various Laplace operators, for example the de Rham Laplacian on forms
and the conformal Laplacian on functions. For a general Laplace type operator
we consider its spectral zeta function Z(s). For a fixed value of s we
calculate the hessian of Z(s) with respect to the metric and show that it is
given by a pseudodifferential operator T(s)=U(s)+V(s) where U(s) is
polyhomogeneous of degree n-2s and V(s) is polyhomogeneous of degree 2. The
operators U(s) and V(s) are meromorphic in s. The symbol expansion of U(s) is
computable from the symbol of the Laplacian and we give an explicit formula for
the principal symbol. Our analysis extends to describe the hessian of the kth
order derivative of Z(s) with respect to s, in particular to the hessian of
Z'(0).
