We have measured the specific heat of an S = 1/2 antiferromagnetic alternating Heisenberg chain pentafulorophenyl nitronyl nitroxide in magnetic fields up to H > H(C2). This compound has a field-induced magnetic ordered (FIMO) phase between H(C1) and H(C2). Characteristic behaviors are observed depending on the magnetic field up to above H(C2) outside of the H-T boundary for the FIMO. The temperature and field dependence of the specific heat are qualitatively in good agreement with the theoretical calculation on an S = 1/2 two-leg ladder [Wang et al., Phys. Rev. Lett. 84, 5399 (2000)]]. This agreement suggests that the observed behaviors are related with the low-energy excitation in the Tomonaga-Luttinger liquid.