TITLE: What is $e^{- \zeta_{\Delta} '(0)}$ for a $\Delta$ the Laplacian of a manifold?
QUESTION [5 upvotes]: For a connected, finite graph $G$, let $\lambda_1, \ldots, \lambda_n$ denote the nonzero eigenvalues of the graph Laplacian. We define $\zeta_G = \Sigma_{i = 1}^n \lambda_i^s$.
Then Kirkoffs Matrix-Tree theorem can be reformulated as saying that $e^{ - \zeta_G'(0)} / |G| = \tau(G)$, where $\tau$ is the number of trees of the graph $G$. (See here for more: http://www.math.ucsd.edu/~fan/lattice.pdf )
If $M$ is a Riemannian manifold, and $\Delta$ is it's Laplacian, we can define a similar zeta function: $\zeta_{\Delta}$. ( https://en.wikipedia.org/wiki/Minakshisundaram%E2%80%93Pleijel_zeta_function )
Question: Is there a reasonable meaning that can be assigned to $e^{- \zeta_{\Delta}'(0)} / Vol(M)$? 
I think (but am not sure) that this quantity doesn't make much sense as stated. The reason is because:
$\zeta_{\Delta}(s) \sim (4 \pi s)^{-n/2} \Sigma_{m = 0}^{\infty} a_m s^m$ (see application one on the wikipedia page) , which has no derivative at zero. 
Thank you!

REPLY [3 votes]: If I am not mistaken, this cannot be true in general. For example, one may scale the metric for $\mathbb{S}^{1}$ proportionally and the determinant of the Laplacian remains the same independent of the length of the circle. For dimension 2, the determinant of the Laplacian obtains its maximum when $M$ has constant sectional curvature. If we rescale the metric proportionally at all places, similar phenomenon happens. One may work with a sphere or a torus as $\mathbb{R}^{2}/L$ with $L$ being a lattice to check explicitly how the $\zeta$-function behaves under dilation.
There is (allegedly) a related conjecture by Nicolas Bergeron, Mehmet Haluk Sengun, Akshay Venkatesh, which claims analytic torsion grows exponentially with respect to the covering of 3-manifolds. I am not entirely sure whether the statement is true (I heard people found counter-examples and the conjecture has to be modified). I think the motivation of $\tau(S)$ may be from elsewhere (like Weyl's paper).