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Visualize Complex Analytic Structure of Prime Zeta Function
PrimeZeta can be analytically continued to 0<Re(s)≤1, it has infinitely many branch points located at where n is a square free natural number.

Click for copyable input
LogLinearPlot[{Re[PrimeZetaP[s]], Im[PrimeZetaP[s]]}, {s, .01, 1.2}, 

 AxesLabel -> {s, P[s]}, PlotStyle -> {Orange, Green}, 

 Epilog -> {Red, 

   Point[{Log[#], 0} & /@ (1/Select[Range[100], SquareFreeQ])]}, 

 PlotRange -> All]