# How come $f(0) = 0$ in $\mathbb C/L$?

How come $f(0) = 0$ in $\mathbb C/L$? Does anyone know it? Your help will be appreciated. This is taken from the text "Rational Points on Elliptic Curves" by Tate and Silverman.

Note that $$f(z_1+ z_2) - f(z_1) - f(z_2) \in L \text{ for all z_1, z_2 close to 0}$$ Now $0$ is (very) close to $0$, hence letting $z_1 = z_2 = 0$, we get: $$f(0) - 2f(0) = -f(0) \in L \iff f(0) \in L$$ As $L$ mapsto $0$ in $\mathbb C/L$ we have $f(0) = 0$ in $\mathbb C/L$.