We simulate directed site percolation on two lattices with 4 spatial and 1 time-like dimensions (simple and body-centered hypercubic in space) with the standard single cluster spreading scheme. For efficiency, the code uses the same ingredients (hashing, histogram re-weighing, and improved estimators) as described in Phys. Rev. E 67, 036101 (2003). Apart from providing the most precise estimates for p_c on these lattices, we provide a detailed comparison with the logarithmic corrections

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... ed by Janssen and Stenull [Phys. Rev. E 69, 016125 (2004)]. Fits with the leading logarithmic terms alone would give estimates of the powers of these logarithms which are too big by typically 50 measured quantities (the average number of sites wetted at time t, their average distance from the seed, and the probability of cluster survival) can be fitted nearly perfectly. But these fits would not be mutually consistent. With a consistent set of fit parameters, one obtains still much improvement over the leading log - approximation. In particular we show that there is one combination of these three observables which seems completely free of logarithmic terms.