We generalize an algorithm used widely in the configuration model such that power-law degree sequences with the degree exponent λ and the number of links per node K controllable independently may be generated. It yields the degree distribution in a different form from that of the static model or under random removal of links while sharing the same λ and K. With this generalized power-law degree distribution, the critical point K c for the appearance of the giant component remains zero not only for λ ≤ 3 but also for 3 < λ < λ l ≃ 3.81. This is contrasted with K c = 0 only for λ ≤ 3 in the static model and under random link removal. The critical exponents and the cluster-size distribution for λ < λ l are also different from known results. By analyzing the moments and the generating function of the degree distribution and comparison with those of other models, we show that the asymptotic behavior and the degree exponent may not be the only properties of the degree distribution relevant to the critical phenomena but that its whole functional form can be relevant. These results can be useful in designing and assessing the structure and robustness of networked systems.