are expressions for volume elements on the discrete moduli spaces associated with the Laplacians, we expect that there are deeper things to be learned from a further study of such expressions.

In the next section we tackle the continuation for arbitrary $p$. The analytic continuation for a general value of $p$ naturally divides into two cases: $p$ odd and $p$ even; in fact we shall see below that the case for $p$ even further divides into two subcases which correspond to $p=0,2\,{\rm mod\,}4$. Due to the size of the expressions it is now much more convenient to continue $\tau_+(p,s)$ and $\tau_-(p,s)$ separately and then combine them into expressions for the torsion. We deal first with $\tau_+(p,s)$ and begin with $p$ odd. Let us recall that $\tau_+(p,s)$ is given by $$\tau_+(p,s)=\sum_{n=1}^\infty
{2(n+1)\left\langle\chi^{n/2}
\right\rangle_p\over \left\{n(n+2)\right\}^s}
\no$$

Reference to the general character formula shows that we must resolve $n$ into its conjugacy classes mod $p$ by writing $$n=pk-j,\;\,j=0,\ldots,(p-1)\no$$ and that we must distinguish the two parities of $j$. To implement these requirements we set $$p=2r+1,\qquad\hbox{and parametrise }
\quad\cases{j& odd by $j=2l+1$, $l=0,1,\ldots,(r-1)$\cr
            j& even by $j=2l$, $l=0,1,\ldots,r$\cr}\no$$ This gives $$\eqalign{\tau_+(p,s)&=2\sum_{j=0}^{p-1}\sum_{k=1}^\infty
{(pk-j+1)\left\langle\chi^{(pk-j)/2}\right\rangle_p
                          \over\left\{(pk-j)(pk-j+2)\right\}^s}\cr
                       &=2\sum_{k=1}^\infty
                      \left[\sum_{l=0}^{r-1}{(pk-2l)
                         \left\langle\chi^{(pk-2l-1)/2}\right\rangle_p
                          \over\left\{(pk-2l-1)(pk-2l+1)\right\}^s}
                        +\sum_{l=0}^r{(pk-2l+1)
                         \left\langle\chi^{(pk-2l)/2}\right\rangle_p
                       \over\left\{(pk-2l)(pk-2l+2)\right\}^s}\right]\cr}\no$$ Then when we use 