  We initiate the computation of the 2-loop quantum AdS_5 x S^5 string
corrections on the example of a certain string configuration in S^5 related by
an analytic continuation to a folded rotating string in AdS_5 in the ``long
string'' limit. The 2-loop term in the energy of the latter should represent
the subleading strong-coupling correction to the cusp anomalous dimension and
thus provide a further check of recent conjectures about the exact structure of
the Bethe ansatz underlying the AdS/CFT duality. We use the conformal gauge and
several choices of the \kappa-symmetry gauge. While we are unable to verify the
cancellation of 2d UV divergences we compute the bosonic contribution to the
effective action and also determine the non-trivial finite part of the
fermionic contribution. Both the bosonic and the fermionic contributions to the
string energy happen to be proportional to the Catalan's constant. The
resulting value for 2-loop superstring prediction for the subleading
coefficient a_2 in the scaling function matches the numerical value found in
hep-th/0611135 from the BES equation.
