I am looking for the best known constant in the boundedness result of the fractional Riesz integral. In particular, I am interested in the dependence on the dimension $d$ and on the parameter $\alpha<d$. Let me rephrase the result I have in mind: for all $0<\alpha <d$, for all $p \in (1,d/\alpha)$ and for all $f \in L^p(\mathbb{R}^d,dx)$, \begin{align*} \|I^\alpha(f)\|_{L^q(\mathbb{R}^d,dx)} \leq C_{\alpha,d,p} \|f\|_{L^p(\mathbb{R}^d,dx)}, \end{align*} where $q$ is given by \begin{align*} \frac{1}{q} = \frac{1}{p}- \frac{\alpha}{d}, \end{align*} and where $I^\alpha$ is the Riesz potential operator (fractional integral). Note that I am aware of Lieb's result regarding the best constant in the fractional Hardy-Littlewood-Sobolev inequality.