Why does the PoissonDistribution not plot around its mean for moderate large numbers?

I plotted the Poisson Distribution for various mean values (plot label and vertical line). The plots does not peak around the mean as the mean increase. Any advice to get the right plot, or at least the right distribution values?

Same when plotting the log of the distribution -s + n Log[s] - LogGamma[n + 1]

Table[ListPlot[Table[PDF[PoissonDistribution[s], n], {n, Round[Max[s - 8 Sqrt[s], 0]], Round[s + 8 Sqrt[s]]}], PlotLabel -> s, GridLines -> {{s}, None}], {s, {50, 60, 70, 80, 100, 150}}]

• Also try DiscretePlot, for example: Table[DiscretePlot[ PDF[PoissonDistribution[s], n], {n, Round[Max[s - 8 Sqrt[s], 0]], Round[s + 8 Sqrt[s]]}, PlotLabel -> s, GridLines -> {{s}, None}], {s, {50, 60, 70, 80, 100, 150}}] – becko Jun 11 '19 at 8:41

Table[ListPlot[