Is this identity or an equivalent one already referenced in the litterature? In particular, is it even true?


$\LARGE{\frac{\left ( -1 ; e^{-4\pi} \right) ^2_{\infty}}{\left ( e^{-2\pi} ; e^{-2\pi} \right) ^4_{\infty}} = \frac{32 \pi^3 \left(\sqrt{2}-1 \right )\sqrt{2^\frac{1}{4}+2^\frac{3}{4}}}{\Gamma^4{\left(\frac{1}{4} \right )}} \simeq 4,030103529...}  (2)$


It appears in relation to a <a href="http://mathoverflow.net/questions/91851/elliptic-function-with-constant-real-part-on-the-unit-square-diagonals">particular elliptic function</a>.

Thanks in advance,