  The thermodynamic Bethe Ansatz equations arising in the context of the
$AdS_5/CFT_4$ correspondence exhibit an important difference with respect to
their analogues in relativistic integrable quantum field theories: their
solutions (the Y functions) are not meromorphic functions of the rapidity, but
live on a complicated Riemann surface with an infinite number of branch points
and therefore enjoy a new kind of extended Y-system. In this paper we review
the analytic properties of the TBA solutions, and present new information
coming from their numerical study. An identity allowing to simplify the
equations and the numerical implementation is presented, together with various
plots which highlight the analytic structure of the Y functions.
