  Typical Tsallis' statistical mechanics' quantifiers like the partition
function and the mean energy exhibit poles. We are speaking of the partition
function ${\cal Z}$ and the mean energy $<{\cal U}>$. The poles appear for
distinctive values of Tsallis' characteristic real parameter $q$, at a
numerable set of rational numbers of the $q-$line. These poles are dealt with
dimensional regularization resources. The physical effects of these poles on
the specific heats are studied here for the two-body classical gravitation
potential.
