By a similar idea for constructing Milnor's gamma functions, we study
``higher depth determinants'' of the Laplacian on a compact Riemann surface of
genus greater than one. We prove that, as a generalization of the determinant
expression of the Selberg zeta function, this higher depth determinant can be
expressed as a product of multiple gamma functions and what we call a
Milnor-Selberg zeta function. Moreover, it is shown that the Milnor-Selberg
zeta function admits an analytic continuation, a functional equation and,
remarkably, has an Euler product.