Playing with the Weierstrass $\wp$-function of the hexagonal (or triangular) lattice $\mathbb{T}$,
$$
\wp'(z)^2 = 4 \wp(z)^3 - 1,
$$
I noticed that the zeros of $\wp'(z) + \sqrt{3}$ are
$$
\frac{\varpi}{3} + \omega,
\quad
e^{\pi i/3} \frac{2\varpi}{3} + \omega,
\quad
e^{2\pi i/3} \frac{\varpi}{3} + \omega,
\qquad
\omega \in \mathbb{T}
$$
(clearly they are all simple), where $\varpi$ is the real period of $\wp$ given by
$$
\varpi = \int_{4^{-1/3}}^{\infty} \frac{dx}{\sqrt{x^3 - 1/4}} = \frac{1}{2 \pi}\, \Gamma(1/3)^3
$$
(the zeros of $\wp'(z) - \sqrt{3}$ can be also determined).
Since
$$
\wp'(z)^2 - 3 = 4 \left[\wp(z)^3 - 1\right]
$$
and $\wp(x)$ is real for $0 < x < \varpi$, we also get
$$
\wp\left(\pm\frac{\varpi}{3}\right) = 1
$$
or, equivalently,
$$
\int_1^{\infty} \frac{dx}{\sqrt{4x^3 - 1}} = \int_{\wp\left(\frac{\varpi}{3}\right)}^{\infty} \frac{dx}{\sqrt{4x^3 - 1}}
= \frac{\varpi}{3}
= \frac{1}{6 \pi}\, \Gamma(1/3)^3
$$
(the detailed calculations can be found in arXiv:2105.04307).

Since I am not an expert in elliptic functions, my question is whether the above facts are known.