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.
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