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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.

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What/who is Sloane? – user12205 Oct 22 '11 at 15:14
@Jeroen: Neil Sloane is OEIS's maintainer. – J. M. Oct 22 '11 at 15:18
If you need a pile of these coefficients, Wolfram Alpha can handle that. – J. M. Oct 22 '11 at 15:30
up vote 3 down vote accepted

They are Jacobi theta functions (with $z=0$)

$$\vartheta_2(q)=\sum_{n=-\infty}^{\infty} q^{(n+1/2)^2}, \qquad \vartheta_3(q)=\sum_{n=-\infty}^{\infty} q^{n^2}$$

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To elaborate a bit: there are four kinds. The second and the third one are what figure into the expression of the modular lambda function. – J. M. Oct 22 '11 at 15:28

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