S=1/2;
J1=10;

NSolve[S+1/2-(1/(2 (π^3)))NIntegrate[((Coth[((6 J1 S (Sqrt[1-(1/9) (((Cos[x]+Cos[y]+Cos[ z]))^2)]))/(2 T1))])/(Sqrt[1-(1/9) (((Cos[x]+Cos[y]+Cos[z]))^2)]))==0,{x,0,π},{y,0,π},{z,0,π}]]