The high-field polynomials L_n(u) (or equivalently the Mayer b_n coefficients) for the spin S=¹/₂ Ising model with nearest-neighbour ferromagnetic interactions (or equivalently the simple lattice-gas) have been studied for a variety of two and three dimensional lattices. It is found that (a) the leading zero of L_n(u) on the positive real u axis always corresponds to a temperature T₁ greater than the critical temperature T_c, and approaches T_c as n to infinity like 1/n¹ Delta /, where Delta is the usual gap exponent. In addition, (b), L_n(u) appears to have exactly n-1 zeros in the physical interval (u_c,1) corresponding to T_c<T< infinity .