I found a "duplicate" question: Entire function having zeros at $m+in$, and I went through some wiki pages of Weierstrass sigma function, but they don't seem to have constructions.
So basically I'm quite confused with the exponent. We can begin with $$f(z) = z\prod_{\omega\in\Lambda^*}\left(1-\frac{z}{\omega}\right)\exp(\dots), \quad \Lambda=\Lambda^*\cup\{0\}=\{m+in\} ,$$ also notice that $$\log\left(1-\frac{z}{\omega}\right) = -\frac{z}{\omega} - \frac{z^2}{2\omega^2} - \dots$$ So why does Weierstrass sigma function put two terms in exponential to get rid of the first two terms? It seems one term is ok to kill all exponent that $\leq 1$.
Is Weierstrass sigma function the "best" construction though?
The reason for the extra factors in the definition of the Weierstrass sigma function is to ensure the product converges absolutely.
"Best construction" is subjective of course, but how about $$(\sin\pi z)\prod_{n=1}^\infty(1-e^{-2n\pi}e^{2\pi iz}) (1-e^{-2n\pi}e^{-2\pi iz})?$$