  In the determination of alpha_s from tau decays several different moments of
the hadronic spectral functions have been used. In a recent work, we performed
an analysis of their perturbative behaviour under two different assumptions for
the higher-order coefficients of the Adler function. We showed that the various
moments can be divided in a small number of classes. We concluded that some of
the moments commonly employed in alpha_s extractions should be avoided due to
their bad perturbative behaviour. Furthermore, for the moments that have a good
perturbative behaviour, and under reasonable assumptions for the higher-order
behaviour of the Adler function, fixed-order perturbation theory (FOPT)
provides the superior framework for the renormalization group improvement. Here
we discuss an extension of this analysis where we consider the perturbative
series for values of the hadronic invariant mass squared s_0 < m_\tau^2. Our
conclusions are not altered within a reasonable s_0 window.
