  For $n$ a nonnegative integer, we consider the $n$-Laplacian $\Delta_n$
acting on the space of $n$-differentials on a confinite Riemann surface $X$
which has ramification points. The trace formula for the resolvent kernel is
developed along the line $\grave{\text{a}}$ la Selberg. Using the trace
formula, we compute the regularized determinant of $\Delta_n+s(s+2n-1)$, from
which we deduce the regularized determinant of $\Delta_n$, denoted by
$\det\!'\Delta_n$. Taking into account the contribution from the absolutely
continuous spectrum, when $n\geq 2$, $\det\!'\Delta_n$ is equal to a constant
$\mathcal{C}_n$ times $Z(n)$, the value of the Selberg zeta function of $X$ at
integer $n$. When $n=0$ or $n=1$, $Z(n)$ is replaced by the leading coefficient
of the Taylor expansion of $Z(s)$ around $s=0$ and $s=1$ respectively. The
constants $\mathcal{C}_n$ are calculated explicitly. They depend on the genus,
the number of cusps, as well as the ramification indices, but is independent of
the moduli parameters.
