From this paper:

  RieszD[\[Alpha]_, f_, x_, opts___] := -1/(2*Cos[\[Alpha]*Pi/2])*1/Gamma[\[Alpha]]*(Integrate[(x - t)^(-\[Alpha] - 1) (f /. x -> t), {t, -Infinity, x}, opts, 
  GenerateConditions -> False] + Integrate[(t - x)^(-\[Alpha] - 1) (f /. x -> t), {t, x, Infinity},opts, GenerateConditions -> False])
  RieszD[\[Alpha]_?Positive, f_, x_, opts___] := Module[{m = Ceiling[\[Alpha]]}, D[RieszD[-(m - \[Alpha]), f, x, opts], {x, m}]]

  f[x_] := Exp[-x]
  RieszD[1/2, f[x], x, Assumptions -> x > 0]
  (* -(((1/2 - I/2) E^-x)/Sqrt[2]) *)

  f1[x_] := Sin[x]
  RieszD[1/2, f1[x], x]
  (* (Sqrt[\[Pi]/2] Sqrt[1/x] Sqrt[x] (Cos[x] - Sin[x]) + Sqrt[\[Pi]/2] (Cos[x] + Sin[x]))/(2 Sqrt[2 \[Pi]]) *)

I'm don't know is my code right, because I have not found any examples to check it out?