In[835]:= EllipticK[0] Out[835]=π/2 so small arguments give π approximations:

From Im[EllipticK[(((1 - 5^(1/4))^8*(Sqrt[5] + 1)^16)/524288)]== (((2*5^(1/4) + 3)*(1/2 - I/10)*(Sqrt[Pi])^3)/((Gamma[3/4])^2*Sqrt[2]))]

In[832]:= N[(50*Gamma[3/4]^4)/(3 + 2*5^(1/4))^2] Out[832]= 3.1415964411401665 or In[834]:= EllipticK[(((2^(1/4) - 1)^4)/((2^(1/4) + 1)^4))] == (((2^(1/4) + 1)^2*(Sqrt[Pi])^3)/(8*(Gamma[3/4])^2*Sqrt[2])) or EllipticK[-(((1 - 5^(1/4))^16*(Sqrt[5] + 1)^8)/65536)]== (((2*5^(1/4) + 3)*(Sqrt[Pi])^3)/(10*(Gamma[3/4])^2*Sqrt[2])) (Aproximating π in terms of π^(3/2) isn't cheating!) Stay tuned for an interesting way to find similar K valuations. --rwg