  In the present note we study absolute linear Harbourne constants. These are
invariants which were introduced in order to relate the lower bounds on the
selfintersection of negative curves on birationally equivalent surfaces to the
complexity of the birational map between them. We provide various lower and
upper bounds on Harbourne constants and give their values for the number of
lines $s$ of the form $p^{2r}+p^r+1$ for any prime number $p$ and also for all
values of $s$ up to $31$. This extends considerably earlier results of the
third author.
