  We have studied numerically the Lee-Yang singularities of the four
dimensional Ising model at criticality, which is believed to be in the same
universality class as the $\phi_4^4$ scalar field theory. We have focused in
the numerical characterization of the logarithmic corrections to the scaling of
the zeros of the partition function and its cumulative probability
distribution, finding a very good agreement with the predictions of the
renormalization group computation on the $\phi_4^4$ scalar field theory. We
have found that this agreement improves much more with the order of the
Lee-Yang zeros. To obtain these results, we have extended a previous study [R.
Kenna and C. B. Lang, Nucl. Phys. {\bf B393} 461 (1993)] in which were computed
numerically the first two zeros for $L\le 24$ lattices, to the computation of
the first four zeros for $L\le 64$ lattices.
