  Context. Rotation is known to affect the nucleosynthesis of light elements in
massive stars, mainly by rotation-induced mixing. In particular, rotation
boosts the primary nitrogen production. Models of rotating stars are able to
reproduce the nitrogen observed in low-Z halo stars. Aims. Here we present the
first grid of stellar models for rotating massive stars at low Z, where a full
s-process network is used to study the impact of rotation-induced mixing on the
nucleosynthesis of heavy elements. Methods. We used the Geneva stellar
evolution code that includes an enlarged reaction network with nuclear species
up to bismuth to calculate 25 M$_\odot$ models at three different Z and with
different initial rotation rates. Results. First, we confirm that
rotation-induced mixing leads to a production of primary $^{22}$Ne, which is
the main neutron source for the s process in massive stars. Therefore rotation
boosts the s process in massive stars at all Z. Second, the neutron-to-seed
ratio increases with decreasing Z in models including rotation, which leads to
the complete consumption of all iron seeds at Z < 1e-3 by the end of core
He-burning. Thus at low Z, the iron seeds are the main limitation for this
boosted s process. Third, as Z decreases, the production of elements up to the
Ba peak increases at the expense of the elements of the Sr peak. We studied the
impact of the initial rotation rate and of the uncertain
$^{17}$O$(\alpha,\gamma)$ rate (which strongly affects the neutron poison
strength of $^{16}$O) on our results. This study shows that rotating models can
produce significant amounts of elements up to Ba over a wide range of Z.
Fourth, compared to the He-core, the primary $^{22}$Ne production in the
He-shell is even higher (> 1% in mass fraction at all Z), which could open the
door for an explosive neutron capture nucleosynthesis in the He-shell, with a
primary neutron source.
