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How come $f(0) = 0$ in 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.

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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 $\mathbb C/L$.

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