HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, z] == 
 (ArcSin[Sqrt[z]] (Pi^2 - 2 ArcSin[Sqrt[z]] (ArcSin[Sqrt[z]] + 3 I Log[1 + E^(-2 I ArcCos[Sqrt[z]])])) + 
  3 I PolyLog[3, E^(2 I ArcSin[Sqrt[z]])] + 6 I PolyLog[3, 1 + E^(-2 I ArcCos[Sqrt[z]])] - 3 I Zeta[3])/(12 Sqrt[z])