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The restriction to <math>\operatorname{Re}\tau=0</math><ref group="note">This was done because the 8th root makes discontinuities in the complex plane.</ref> can be dropped if we [[Analytic continuation|analytically extend]] <math>u</math> and <math>v</math> by
:<math>u=\sqrt{2}e^{p\pi i\tau/8}\prod_{k=1}^\infty \frac{1+e^{2kp\pi i\tau}}{1+e^{(2k-1)p\pi i\tau}},\quad v=\sqrt{2}e^{\pi i\tau/8}\prod_{k=1}^\infty \frac{1+e^{2k\pi i\tau}}{1+e^{(2k-1)\pi i\tau}}.</math>
Since <math>\lambda(i)=1/2</math>, the modular equations can be used to give [[Algebraic number|algebraic values]] of <math>\lambda(pi)</math> for any prime <math>p</math>. For any odd <math>n</math>, the algebraic values of <math>\lambda(ni)</math> are given by<ref>{{Cite book |last1=Jacobi |first1=Carl Gustav Jacob |title=Fundamenta nova theoriae functionum ellipticarum|year=1827}} p. 42</ref><ref group="note"><math>\operatorname{sl}a\varpi</math> is algebraic for every <math>a\in\mathbb{Q}.</math></ref>
:<math>\lambda (ni)=\frac{1}{2^n}\prod_{k=1}^n \left(1-\operatorname{sl}^2\frac{k\varpi}{n}\right)^2</math>
where <math>\operatorname{sl}</math> is the [[Lemniscate elliptic functions|lemniscate sine]] and <math>\varpi</math> is the [[Lemniscate elliptic functions#Lemniscate constant|lemiscate constant]].
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