I think they're mislabelled and misdescribed a bit, and what those graphs actually show is what you get if you (1) compute an approximation to J using only a li

I think they're mislabelled and misdescribed a bit, and what those graphs actually show is what you get if you (1) compute an approximation to J using only a limited number of Riemann zeros, and then (2) use the inversion formula to compute what pi(x) would be if #1 had given you the exact value of J. So the y-axis shouldn't be labelled "J(x)", it should be labelled something more like "pi(x)". Caution: I haven't actually done the computation and checked. [EDITED to add:] Now I have, or at least one of them, for the "first 35 roots" one. I took "first 35 roots" to mean "first 35 with positive imaginary part, plus their complex conjugates". I got a graph that was much wigglier, and a much closer match to pi(x) than the one in the article. So then I thought maybe it was meant to be 35 roots in total -- though you really do want to take those conjugates in pairs, so the odd number is strange. Anyway, I tried with the first 17 pairs: still much too wiggly. With the first five pairs of zeros, I get a good (but not perfect) match for the graph in the article.