It is known that perturbative invariants of rational homology 3‐spheres can be constructed by using arithmetic perturbative expansion of quantum invariants of them. However, we could not make arithmetic perturbative expansion of quantum invariants for 3‐manifolds with positive Betti numbers by the same method. In this paper, we explain how to make arithmetic perturbative expansion of quantum SO.3/ invariants of 3‐manifolds with the first Betti number 1. Further, motivated by this expansion, we construct perturbative invariants of such 3‐manifolds. We show some properties of the perturbative invariants, which imply that their coefficients are independent invariants. 57M27 In the late 1980s, Witten [44] proposed topological invariants of a closed 3‐manifold M for a simple compact Lie group G , what we call quantum G invariant, which is formally presented by a path integral whose Lagrangian is the Chern‐Simons functional of G connections on M . There are two approaches to obtain mathematically rigorous information from a path integral: the operator formalism and the perturbative expansion. Motivated by the operator formalism of the Chern‐Simons path integral, Reshetikhin and Turaev [34] gave the first rigorous mathematical construction of quantum invariants, as linear sums of quantum invariants of framed links. After that, rigorous constructions of quantum invariants were obtained by various approaches; in particular, Kirby and Melvin [14] constructed the quantum SO.3/ invariant, which we denote by SO.3/ r .M/;