  Let $M$ be a finite volume, non-compact hyperbolic Riemann surface, possibly
with elliptic fixed points, and let $\chi$ denote a finite dimensional unitary
representation of the fundamental group of $M$. Let $\Delta$ denote the
hyperbolic Laplacian which acts on smooth sections of the flat bundle over $M$
associated to $\chi$. From the spectral theory of $\Delta$, there are three
distinct sequences of numbers: The first coming from the eigenvalues of $L^{2}$
eigenfunctions, the second coming from resonances associated to the continuous
spectrum, and the third being the set of negative integers. Using these
sequences of spectral data, we employ the super-zeta approach to regularization
and introduce two super-zeta functions, $\Z_-(s,z)$ and $\Z_+(s,z)$ that encode
the spectrum of $\Delta$ in such a way that they can be used to define the
regularized determinant of $\Delta-z(1-z)I$. The resulting formula for the
regularized determinant of $\Delta-z(1-z)I$ in terms of the Selberg zeta
function, see Theorem 5.3, encodes the symmetry $z\leftrightarrow 1-z$, which
could not be seen in previous works, due to a different definition of the
regularized determinant.
