Evaluate $\int_{\Bbb S^n}\int_{\Bbb S^n}|s-t|^{-\lambda}\rm dsdt$

$J(s)=\int_{\mathbb{S}^n}|s-t|^{-\lambda}{\rm d}t$ doesn't (actually) depend on $s\in\mathbb{S}^n$, so that $$\int_{\mathbb{S}^n}\int_{\mathbb{S}^n}\frac{{\rm d}s\,{\rm d}t}{|s-t|^\lambda}=S_n J(s_0),$$ where $S_n=2\pi^{n/2}/\Gamma(n/2)$ is the measure of $\mathbb{S}^n$, and $s_0=(1,0,\dots,0)$ say. Now, for $t=(t_1,t_2,\dots,t_n)$ with $t_1=x\in(-1,1)$ fixed, we have $|s_0-t|^2=2(1-x)$ and $t_2^2+\dots+t_n^2=1-x^2$; thus, $$\int_{\mathbb{S}^n}\int_{\mathbb{S}^n}\frac{{\rm d}s\,{\rm d}t}{|s-t|^\lambda}=S_n S_{n-1}\int_{-1}^1\frac{(1-x^2)^{n/2-1}}{\big(2(1-x)\big)^{\lambda/2}}\,{\rm d}x,$$ and the last integral is of beta type (after $x=1-2y$). The result is $$\int_{\mathbb{S}^n}\int_{\mathbb{S}^n}\frac{{\rm d}s\,{\rm d}t}{|s-t|^\lambda}=\frac{2^{n-\lambda+1}\pi^{n-1/2}\Gamma\left(\frac{n-\lambda}{2}\right)}{\Gamma\left(\frac{n-1}{2}\right)\Gamma\left(n-\frac{\lambda}{2}\right)}.$$