Let \mathfrak{p} be the defining ideal of the monomial curve ${\mathcal {C}}(2q+1, 2q+1+m, 2q+1+2m)$ in the affine space $\mathbb {A}_{k}^{3}$ parameterised by $(x^{2q+1}, x^{2q+1+m}, x^{2q+1+2m})$, where $\mathbb {A}_{k}^{3}$. In this paper we compute the resurgence of \mathfrak{p}, the Waldschmidt constant of $\mathfrak {p}$ and the Castelnuovo-Mumford regularity of the symbolic powers of $\mathfrak {p} $.