  We study six-point gluon scattering amplitudes in N=4 super Yang-Mills theory
at strong coupling by investigating the thermodynamic Bethe ansatz equations of
the underlying Z_4-symmetric integrable model both analytically and
numerically. By the conformal field theory (CFT) perturbation, we compute the
free energy part of the remainder function with generic chemical potential near
the CFT/small mass limit. Combining this with the expansion of the Y-functions,
we obtain the remainder function near the small mass limit up to a function of
the chemical potential, which can be evaluated numerically. We also find the
leading corrections to the remainder function near the large mass limit. We
confirm that these results are in good agreement with numerical computations.
