  We study the thermodynamics of the antiperiodic XXZ chain with anisotropy
parameter {\eta}=i{\pi}/3 by means of the t-W method. We parameterize the
eigenvalues of both the transfer matrix and the corresponding fused transfer
matrix by their zero points instead of Bethe roots. Based on the patterns of
the zero points distribution and the reconstructed entropy, we obtain the
nonlinear integral equations (NLIEs) describing the thermodynamics of the model
and compute its free energy at a finite temperature.
