= Spectral zeta function
{title2=$\zeta_L(s)=\sum_{\lambda_j>0}\lambda_j^{-s}$}
For a nonnegative elliptic operator $L$ on a <closed manifold>, sum its positive <eigenvalues> with multiplicity. The defining series converges in a right half-plane. If $n_0=\dim\ker L$ and $K(t)=\operatorname{Tr}(e^{-tL})$ is the <heat trace>, then
$$
\zeta_L(s)=\frac1{\Gamma(s)}\int_0^\infty t^{s-1}(K(t)-n_0)\,dt.
$$
Subtracting finitely many terms of the short-time <heat kernel expansion> gives a <meromorphic continuation>. Zero eigenvalues must be excluded: assigning them a nonexistent inverse power does not define the spectral zeta function.
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