Consider the Weierstrass $\wp$ elliptic function $\wp(z, g_2, g_3)$ with the invariants $g_2\in\mathbb{R}$ and $g_3\in\mathbb{R}$:
$$\wp'(z)^2 = 4\wp(z)^3 - g_2 \wp(z) - g_3$$
According to Wikipedia when $\Delta=g_ 2^3-27g_3^2 > 0$, we have three real roots $e_1 > e_2 > e_3$ and $\wp(z, g_2, g_3)\in\mathbb{R}$.
My question is: in that case (where all values are real, including $z$), how to compute the period $w\in\mathbb{R}$ (or half-period) of $\wp(z, g_2, g_3)$ as a function of $g_2, g_3, e_1, e_2, e_3$ ? (I am searching for an analytical formula in terms of standard functions or special functions available in this list, not in terms of an integral).
Additional note : If understand correctly, the same Wikipedia article seems to say that : $$\frac{w_{1}}{2}=\int_{e_1}^{\infty}\frac{dz}{\sqrt{4z^3-g_2 z-g_3}}$$ but I am not sure that their $w_1$ is the $w$ I am searching for. In any case I would like a formula in terms of special functions as explained above.
Your understanding of the formula is correct. The period is given by a complete elliptic integral of the first kind; there's no escaping it. (At least, you'll be happy to know that the complete elliptic integral appears in your list.)
By a change of variables, you can put your elliptic curve into Legendre form $y^2=x(x-1)(x-\lambda)$; you should have $\lambda = (e_3-e_2)/(e_1-e_2)$. Then you can evaluate the integral numerically using Jacobi's form of the complete elliptic integral, with $k^2=\lambda$.