In this paper for the fractional Riemann-Liouville integrals \[ P_ rf(x)=\frac{1}{\Gamma (r)}\int^{x}_{0}(x-t)^{r-1}f(t)dt,\quad r>0, \] the weight estimates of the type \[ (*)\quad (\int^{\infty}_{0}| P_ rf(x)u(x)|^ pdx)^{1/p}\leq C(\int^{\infty}_{0}| f(x)v(x)|^ pdx)^{1/p} \] are studied. More precisely, in the case \(r>0\) and \(p=2\) the problem of complete description of weights u and v for which the estimation (*) is fulfilled, is studied. In particular, the following theorem is proved. Theorem 1. Let \(r\geq 1\). The estimation (*) is valid if and only if \[ A_{r-1,2}=\max_{\gamma =0,1}\quad \sup_{t>0}A_{r- 1,2,\gamma}(t)0}\Gamma^{-1}(r)(\int^{\infty}_{0}(x- t)^{2(r-1)(1-\gamma)}| u(x)|^ 2dx)^{1/2}\times \] \[ \times (\int^{t}_{0}(t-x)^{2(r-1)\gamma}| v(x)|^{- 2}dx)^{1/2}<\infty. \] Moreover, if C is the smallest constant in (*) (in the case \(p=2)\), then \(A_{r-1,2}\leq C\leq \alpha A_{r-1,2,}\) where \(\alpha\) is an absolute positive constant.