Summary

Overall, the $Z(T,D)$ is found by:

$$\begin{eqnarray*} \log(\beta_D) & \approx & -\frac{1}{2} \log(D) + \frac{1}{2} ... ...+ 3 Z_n^{-4})} \quad \mathrm{for~} n=1,2,3 \\ Z & \approx & Z_3 \end{eqnarray*}$$

This approximation has a relative accuracy of $10^{-3}$ or better for

$D \ge 15$	and	$T \ge 7.5$
$D<15$	and	$-14.5 > \log(p) \approx -\frac{1}{2} \log(D) - \log(\beta_D) - \log(T) - \frac{D-1}{2} \log\left(1 + \frac{T^2}{D}\right)$

Outside this valid domain, the probability is always greater than $10^{-14}$ and can therefore be calculated by conventional means.