  We extend the definitions of characters and partition functions to the case
of conformal field theories which contain operators with logarithmic
correlation functions. As an example we consider the theories with central
charge c = c(p,1) = 13-6(p+1/p), the ``border'' of the discrete minimal series.
We show that there is a slightly generalized form of the property of
rationality for such logarithmic theories. In particular, we obtain a
classification of theories with c = c(p,1) which is similar to the A-D-E
classification of c = 1 models.
