  We consider a circular string with spin $S$ in $AdS_5$ wrapped around big
circle of $S^5$ and carrying also momentum $J$. The corresponding N=4 SYM
operator belongs to the SL(2) sector, i.e. has tr$(D^S Z^J)+...$ structure. The
leading large $J$ term in its 1-loop anomalous dimension can be computed using
Bethe ansatz for the SL(2) spin chain and was previously found to match the
leading term in the classical string energy. The string solution is stable at
large $J$, and the Lagrangian for string fluctuations has constant
coefficients, so that the 1-loop string correction to the energy $E_1$ is given
simply by the sum of characteristic frequencies. Curiously, we find that the
leading term in the zero-mode part of $E_1$ is the same as a 1/J correction to
the one-loop anomalous dimension on the gauge theory (spin chain) side that was
found in hep-th/0410105. However, the contribution of non-zero string modes
does not vanish. We also discuss the ``fast string'' expansion of the classical
string action which coincides with the coherent state action of the SL(2) spin
chain at the first order in $\l$, and extend this expansion to higher orders
clarifying the role of the $S^5$ winding number.
