The lambda invariants at CM points

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 $\lambda(\frac{d_1+\sqrt{d_1}}2) -\lambda(\frac{d_2+\sqrt{d_2}}2)$,withthelatterunderthecondition$(d_1, d_2)=1$\mathrm{.}Finally,weusetheseresultstoshowthat$\lambda(\frac{d+\sqrt d}2)$ 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.