  In the paper, we show that $\lambda(z_1) -\lambda(z_2)$, $\lambda(z_1)$ and
$1-\lambda(z_1)$ are all Borcherds products in $X(2) \times X(2)$. 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.
