Closed geodesics on hyperbolic surfaces with few intersections

We prove that, if a closed geodesic $\mathrm{\Gamma }$ on a complete finite type hyperbolic surface has at least 2 self-intersections, then the length of $\mathrm{\Gamma }$ has an lower bound $2\log (5+2\sqrt{6})$, and the lower bound is sharp, attained on a corkscrew geodesic on a thrice punctured sphere.