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Abstract In this paper, we show that $$\lambda (z_1) -\lambda (z_2)$$, $$\lambda (z_1)$$, and $$1-\lambda (z_1)$$ are all Borcherds products on $$X(2) \times X(2)$$. We then use the big CM value formula of Bruinier, Kudla, and Yang to give explicit factorization formulas for the norms of $$\lambda (\frac{d+\sqrt d}2)$$, $$1-\lambda (\frac{d+\sqrt d}2)$$, and $$\lambda (\frac{d_1+\sqrt{d_1}}2) -\lambda (\frac{d_2+\sqrt{d_2}}2)$$, with the latter under the condition $$(d_1, d_2)=1$$. Finally, we use these results to show that $$\lambda (\frac{d+\sqrt d}2)$$ is always an algebraic integer and can be easily used to construct units in the ray class field of $${\mathbb{Q}}(\sqrt{d})$$ of modulus $$2$$. In the process, we also give explicit formulas for a whole family of local Whittaker functions, which are of independent interest.
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