We construct random walks on free products of the form Z 3 * Z d , with d = 5 or 6 which are divergent and not spectrally positive recurrent. We then derive a local limit theorem for these random walks, proving that µ * n (e) ∼ CR −n n −5/3 if d = 5 and µ * n (e) ∼ CR −n n −3/2 log(n) −1/2 if d = 6, where µ * n is the nth convolution power of µ and R is the inverse of the spectral radius of µ. This disproves a result of Candellero and Gilch [7] and a result of the authors of this paper that was stated in a rst version of [11]. This also shows that the classication of local limit theorems on free products of the form Z d 1 * Z d 2 or more generally on relatively hyperbolic groups with respect to virtually abelian subgroups is incomplete.