  We consider the critical non-unitary minimal model {\cal M}(3,5) with
integrable boundaries. We analyze the patterns of zeros of the eigenvalues of
the transfer matrix and then determine the spectrum of the critical theory
through the Thermodynamic Bethe Ansatz (TBA) equations. By solving the TBA
functional equation satisfied by the transfer matrices of the associated A_{4}
RSOS lattice model of Forrester and Baxter in Regime III in the continuum
scaling limit, we derive the integral TBA equations for all excitations in the
(r=1,s=1) sector then determine their corresponding energies. The excitations
are classified in terms of (m,n) systems.
