is now complete.

Evaluating our expressions at $s=0$ we find $$\eqalign{\tau_{+}(2,0)&=8\zeta(-2) +2R(0)
\qquad \tau_{-}(2,0)=-6\zeta(-2)\cr
&{\rm and}\qquad
\tau(2,0)=14\zeta(-2)\cr
}\no$$ Observe that with our continuation $R(0)$ is automatically zero. Thus noting also that $\zeta(-2)=0$, we conclude $$\tau_{+}(2,0)=0\, , \qquad \tau_{-}(2,0)=0\, ,
\qquad and \qquad\tau(2,0)=0\no$$ That $\tau_{\pm}(p,0)=0$ is quite generally true for arbitrary $p$; we shall see this in the next section and this agrees with general considerations for generalised zeta functions of second order operators on compact odd dimensional manifolds.

We can now take the final step which is to differentiate and obtain $\tp(2,0)$ and $\tm(2,0)$, which we denote by $\tp(2)$, $\tm(2)$ respectively, and hence the torsion $T(2)$.

The resulting expressions are $$\tp(2)= 16\zeta '(-2)+2\zeta(0)+2R^\prime(0)$$ $$\tm(2)= -12\zeta '(-2)-2\ln4\zeta(0)$$ and for the torsion $$\ln T(2)={d\tau(2,0)\over ds}=28\zeta^\prime(-2)
+2(1+\ln4)\zeta(0)+2R^\prime(0)\no$$ But $$\zeta(0)=-1/2,\qquad\hbox{ and }\qquad \zeta^\prime(-2)=-
{\zeta(3)\over4\pi^2}\quad\hbox{from the functional relation}\no$$ and by our remark above concerning the motive for our choice of definition for $R(m,s)$ we have $$\eqalign{R^\prime(0)&=
{d\over ds}\sum_m\left. R(m,s)\right\vert_{s=0}\cr
\Rightarrow R^\prime(0)&=\sum_m\left.{d R(m,s)\over ds}\right\vert_{s=0}\cr
                       &=\sum_m\left[4m^2\left\{\ln(4m^2)-
\ln(4m^2-1)\right\}-1\right]=-
\sum_m\left[4m^2\ln(1-1/4m^2)+1\right]\cr}\no$$ Hence $$\eqalign{
\tp(2)&=-{4\over\pi^2}\zeta(3)-1
-2\sum_m\left[4m^2\ln(1-1/4m^2)+1\right]\cr
\tm(2)&={3\over\pi^2}\zeta(3)+2\ln2\qquad {\rm and}\cr
\ln T(2)&=-{7\over\pi^2}\zeta(3)-1-2\ln(2)-
2\sum_m\left[4m^2\ln(1-1/4m^2)+1\right]}\no$$ However the series for $R^\prime(0)$ can be expressed as a trigonometric integral; in fact, as a special case of more general results which will be derived below, we have 