Call $\pi \colon X \to \mathbb{P}^3$ the first blow-up. The normal bundle of $L_2$ in $\mathbb{P^3}$ is $$N_{L_2/ P^3}=\mathbb{O}_{P^3}(1) \oplus \mathbb{O}_{P^3}(1),$$ in fact $L_2$ moves into a family of dimension $4=h^0(N_{L_2/ P^3})$, na
he strict transform $L_2'$ of $L_2$ moves into a family