  We discuss the analytical determination of the location of the Last Stable
Orbit (LSO) in circular general relativistic orbits of two point masses. We use
several different ``resummation methods'' (including new ones) based on the
consideration of gauge-invariant functions, and compare the results they give
at the third post-Newtonian (3PN) approximation of general relativity. Our
treatment is based on the 3PN Hamiltonian of Jaranowski and Sch\"afer. One of
the new methods we introduce is based on the consideration of the (invariant)
function linking the angular momentum and the angular frequency. We also
generalize the ``effective one-body'' approach of Buonanno and Damour by
introducing a non-minimal (i.e. ``non-geodesic'') effective dynamics at the 3PN
level. We find that the location of the LSO sensitively depends on the
(currently unknown) value of the dimensionless quantity $\oms$ which
parametrizes a certain regularization ambiguity of the 3PN dynamics. We find,
however, that all the analytical methods we use numerically agree between
themselves if the value of this parameter is $\oms\simeq-9$. This suggests that
the correct value of $\oms$ is near -9 (the precise value
$\oms^*\equiv-{47/3}+{41/64}\pi^2=-9.3439...$ seems to play a special role). If
this is the case, we then show how to further improve the analytical
determination of various LSO quantities by using a ``Shanks'' transformation to
accelerate the convergence of the successive (already resummed) PN estimates.
