  It is well known that multi-point Seshadri constants for a small number $s$
of points in the projective plane are submaximal. It is predicted by the Nagata
conjecture that their values are maximal for $s\geq 9$ points. Tackling the
problem in the language of valuations one can make sense of $s$ points for any
positive real $s\geq 1$. We show somewhat surprisingly that a Nagata-type
conjecture should be valid for $s\geq 8+1/36$ points and we compute explicitly
all Seshadri constants (expressed here as the asymptotic maximal vanishing
element) for $s\leq 7+1/9$.
