Normalization of the hypervirial distribution function ($C=\frac{(p+1)\Gamma (2p+2)}{2^{(p+5)/2}{\pi }^{5/2}\Gamma (p/2)\Gamma ((3p+3)/2)}$)

In units $G=M=a=1$, velocity integration of $F=C{L}^{p-2}{E}^{(3p+1)/2}$ gives $C=(p+1)\Gamma (2p+2)/[2^{(p+5)/2}{\pi }^{5/2}\Gamma (p/2)\Gamma ((3p+3)/2)]$. The beta function separates the radial-speed and angular integrals. This constant is positive for every $p>0$.