Hessians of spectral zeta functions

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 p-forms and the conformal Laplacian on functions. For a general geometric differential operator of Laplace type with eigenvalues 0 less than or equal to lambda(1) less than or equal to lambda(2) less than or equal to (...), we consider the spectral zeta function Z(s) = Sigma(lambdanot equal0)lambda(j)(-s). The modified zeta function L(s) =Gamma(s) Z (s) / Gamma(s - n/2) is an entire function of s. For a fixed value of s, we calculate the Hessian of L(s) with respect to the metric and show that it is given by a pseudodifferential operator T-s = U-s + Y-s where U-s is polyhomogeneous of degree n - 2s and V-s is polyhomogeneous of degree 2. The operators U-s/ Gamma(n/2 + 1 - s) and V-s / Gamma(n/2 + I - s) are entire in s. The symbol expansion of U-s is computable from the symbol of the Laplacian. Our analysis extends to describing the Hessian of (d/ds)(k) L(s) for an), value of k.