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The Waldschmidt constant $\alphahat(I)$ of a radical ideal $I$ in thecoordinate ring of $\PP^N$ measures (asymptotically) the degree of ahypersurface passing through the set defined by $I$ in $\PP^N$. Nagata'sapproach to the 14th Hilbert Problem was based on computing such constant forthe set of points in $\PP^2$. Since then, these constants drew much attention,but still there are no methods to compute them (except for trivial cases).Therefore the research focuses on looking for accurate bounds for$\alphahat(I)$.  In the paper we deal with $\alphahat(s)$, the Waldschmidt constant for $s$very general lines in $\PP^3$. We prove that $\alphahat(s) \geq\lfloor\sqrt{2s-1}\rfloor$ holds for all $s$, whereas the much stronger bound$\alphahat(s) \geq \lfloor\sqrt{2.5 s}\rfloor$ holds for all $s$ but $s=4$, $7$and $10$. We also provide an algorithm which gives even better bounds for$\alphahat(s)$, very close to the known upper bounds, which are conjecturallyequal to $\alphahat(s)$ for $s$ large enough.

Lower bounds for Waldschmidt constants of generic lines in $\mathbb{P}^3$ and a Chudnovsky-type theorem