Following what has been discussed here and here, which I found by doing some research prior to asking my question, it seems like the error function, defined as $$\text{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{-t^2}\text{d}t,$$ is some intricate object. No special values are known, besides $0$, $\pm\infty$ and $\pm i\infty$. However, what I was wondering is the following :
Is there a known result about rationality (or even algebraicity) of special values of the error function ?
That is, are there some known special values $\alpha>0$ such that $\text{erf}(\alpha)\in\mathbb{Q}$ ? This arose from when I wondered whether we could say something about $\alpha$ if it was such that $$\int_\alpha^{+\infty}e^{-t^2}\text{d}t=q\sqrt{\pi}$$
with $0<q<1$ some rational $-$ which indeed amounts to knowing when $\text{erf}(\alpha)$ is rational.
I known this doesn't seem like a much simpler question to ask than knowing specifically which value it takes at some points, but I perhaps hope that someone knows a bit of an answer to this fact. I am also interested in knowing whether some special values are known to be algebraic rather than rational too !
As a side note, perhaps results like the following may be used in some way ?
Theorem : (Lindemann-Weierstraß) If $a_1,...,a_n$ are algebraic numbers linearly independant over $\mathbb{Q}$, then $e^{a_1},...,e^{a_n}$ are algebraically independant over $\mathbb{Q}$.
