Hi, for the life of me I cannot get the same result as you did for the 𝜈=5/2 case. I get the same overall shape but a factor 16/3 instead of the 3/2 you have:
S = \alpha^2 \frac{2 \sqrt{\pi} \Gamma(\nu + 1/2)(2\nu)^{\nu}}{\Gamma(\nu)\rho^{2\nu}}(...)^{-3} \\ = \alpha^2 \frac{2 \sqrt{\pi} \Gamma(3)5^{5/2}}{\Gamma(5/2)\rho^{5}}(...)^{-3} \\ = \alpha^2 \frac{2 \sqrt{\pi}\cdot 2 \cdot 5^{5/2}}{\frac{3\sqrt{\pi}}{4}\cdot \rho^{5}}(...)^{-3} \\ = \alpha^2 \frac{2 \cdot 2 \cdot 5^{5/2}}{\frac{3}{4}\cdot \rho^{5}}(...)^{-3} \\ = \alpha^2 \frac{16}{3} (\frac{\sqrt{5}}{\rho})^5(...)^{-3}
What do you think, which one is correct?
Thanks!