  We study at strong coupling the scaling function describing the large spin
anomalous dimension of twist two operators in ${\cal N}=4$ super Yang-Mills
theory. In the spirit of AdS/CFT duality, it is possible to extract it from the
string Bethe Ansatz equations in the $\mathfrak{sl}(2)$ sector of the $\ads$
superstring. To this aim, we present a detailed analysis of the Bethe equations
by numerical and analytical methods. We recover several short string
semiclassical results as a check. In the more difficult case of the long string
limit providing the scaling function, we analyze the strong coupling version of
the Eden-Staudacher equation, including the Arutyunov-Frolov-Staudacher phase.
We prove that it admits a unique solution, at least in perturbation theory,
leading to the correct prediction consistent with semiclassical string
calculations.
