  The Waldschmidt constant $\alphahat(I)$ of a radical ideal $I$ in the
coordinate ring of $\PP^N$ measures (asymptotically) the degree of a
hypersurface passing through the set defined by $I$ in $\PP^N$. Nagata's
approach to the 14th Hilbert Problem was based on computing such constant for
the 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 conjecturally
equal to $\alphahat(s)$ for $s$ large enough.
