Looking at Sloane's database, I found a neat formula for the lambda-invariant. Let $q:\tau \mapsto \exp(\pi i \tau)$ on the complex upper-half plane. Then
$\lambda(q) = 16q\;\prod_{k>0} \frac{(1 + q^{2k})^8}{(1 + q^{2k - 1})^8} = \frac{\vartheta_2(q)^4}{\vartheta_3(q)^4}.$
What are those theta functions?
I found this formula here.