N[WeierstrassInvariants[{1/2,I/2}]]

WeierstrassP[1/2,{c,0}]

c=189.07272;
e1=Sqrt[c/4];2Sqrt[2Pi]e1

costa1[u_,v_]:= Re[-WeierstrassZeta[u+I v,{c,0}]+Pi u+Pi^2/(4 e1)+(Pi/(2 e1)) (WeierstrassZeta[u+I v-1/2,{c,0}]-WeierstrassZeta[u+I v-I/2,{c,0}])]

costa2[u_,v_]:=Re[-I*WeierstrassZeta[u+I v,{c,0}]+Pi v+Pi^2/(4 e1)-(Pi/(2 e1))*(I*WeierstrassZeta[u+I v-1/2,{c,0}]-I*WeierstrassZeta[u+I v-I/2,{c,0}])]

costa3[u_,v_]:=(Sqrt[2 Pi]/2) Log[Abs[(WeierstrassP[u+I v,{c,0}]-e1)/(WeierstrassP[u+I v,{c,0}]+e1)]]

costa[u_,v_]:={costa1[u,v],costa1[v,u],costa3[u,v]}

ParametricPlot3D[costa[u,v],{u,0.0001,1},{v,0.0001,1},PlotPoints->40,PlotRange->{{-7,7},{-7,7},{-4,4}},AspectRatio->Automatic,Axes->False,Boxed->False,Mesh->None]

Simplify[costa1[3, v] - costa2[v, u]]

ParametricPlot3D[costa[u,v],{u,0.0001,1},{v,0.0001,1},PlotPoints->40,PlotRange->{{-7,7},{-7,7},{-1,1}},AspectRatio->Automatic,Axes->False,Boxed->False,Mesh->None]

Here is another parametrization:(*runtime:5 seconds*)

Costa[z_]:=
Module[{phi1=-2 Sqrt[z] Sqrt[1-z^2] Hypergeometric2F1[1/4,3/2,5/4,z^2]/Sqrt[z^2-1],phi2=-(2/3) z^(3/2) Sqrt[z^2-1] Hypergeometric2F1[3/4,1/2,7/4,z^2]/Sqrt[1-z^2]},Re[{phi2-phi1,I(phi1+phi2),Log[z-1]-Log[z+1]}]/2];
surface=ParametricPlot3D[Costa[Sqrt[Exp[r-I theta]+1]],{r,-3.5,6},{theta,-Pi,Pi},PlotPoints->{20,18},Compiled->False][[1]];
<<Graphics`Shapes`;surface={surface,RotateShape[surface,Pi,0,0]};Show[Graphics3D[{surface,RotateShape[surface,Pi/2,Pi,0]}]]

Richmond[n_,z_]:={-1/(2z)-z^(2n+1)/(4n+2),-I/(2z)+I z^(2n+1)/(4n+2),z^n/n};
ParametricPlot3D[Re[Richmond[5,r Exp[I theta]]],{r,0.53,1.187},{theta,0,2Pi},PlotPoints->{25,180},Compiled->False]

n=5;psi1[z_]:=n z^(1/n) Hypergeometric2F1[0.5/n,1+1/n,1+0.5/n,z^2];psi2[z_]:=(n/(2 n-1)) z^(2-1/n) Hypergeometric2F1[1-0.5/n,1-1/n,2-0.5/n,z^2];
rho=Sqrt[4^(1/n)Gamma[1-0.5/n] Gamma[(n+1)/(2 n)]/(Gamma[1+0.5/n] Gamma[(n-1)/(2 n)])];f[z_]:=0.5Re[{psi2[z]/rho-rho psi1[z],I (psi2[z]/rho+rho psi1[z]),Log[(z-1)/(z+1)]}];
surface=ParametricPlot3D[f[Sqrt[Exp[x+I y]+1]],{x,-3,6},{y,1.0*^-6,Pi},PlotPoints->{30,15},Compiled->False,DisplayFunction->Identity][[1]];
<<Graphics`Shapes`;surface={surface,RotateShape[surface,Pi/n,Pi,0]};surface={surface,AffineShape[surface,{1,-1,1}]};
Show[Graphics3D[Table[RotateShape[surface,2Pi i/n,0,0],{i,1,n}]]];