We study a system of coalescing continuous-time random walks starting from every site on $\mathbb{Z}$, where the jump increments lie in the domain of attraction of an $α$-stable distribution with $α\in(0,1]$. We establish sharp asymptotics for the $N$-point correlation function of the system. Our analysis relies on two precise tail estimates for the system density, as well as the non-collision probability of $N$ independent random walks with arbitrary fixed initial configurations. In addition, we derive refined estimates for heavy-tailed random walks, which may be of independent interest.