  From thermodynamics, LSDA+U calculations and exact diagonalization studies of
a five-band Hubbard (Cu 3d, O 2p) model on CuO2 stripes we found that Li2ZrCuO4
(Li2CuZrO4 in traditional notation) is close to a ferromagnetic critical point.
Analyzing its measured susceptibility chi(T) and specific heat cp(T,H) within a
Heisenberg model, we show that the ratio of the 2nd to the 1st neighbor
exchange integrals alpha=-J2/J1 of the order of 0.3, i.e. it is close to the
critical value alpha_c=1/4. Comparing with related chain cuprates we explain
the rather strong field dependence of cp, the monotonous down shift of the peak
of chi(T), and its increase for alpha approaching 1/4 + 0.
