Let $\Delta_{M_1}$ and $\Delta_{M_2}$ be the Laplace-Beltrami operators on two compact and connected Riemannian manifolds $M_1$ and $M_2$ respectively. We define the spectral zeta function (or Minakshisundaram–Pleijel zeta function) by: $$ \zeta_M(s)=\sum_{i=1}^\infty \lambda_i^{-s},\quad Re(s)\gg 0 $$ where {$0<\lambda_1\le\lambda_2\le\lambda_3\le\cdot\cdot\cdot$} are the non-zero eigenvalues of $\Delta_M$.
How can one prove the following?
If $\zeta_{M_1}(s)=\zeta_{M_2}(s)$, then $M_1$ and $M_2$ are isospectral in the sense that the eigenvalues of $\Delta_{M_1}$ and $\Delta_{M_2}$ are identical.