In long-range percolation on $\mathbb{Z}^d$, points $x$ and $y$ are connected by an edge with probability $1-\exp(-β\|x-y\|^{-d-α})$, where $α>0$ is fixed and $β\geq 0$ is a parameter. As $d$ and $α$ vary, the model is conjectured to exhibit eight qualitatively different second-order critical behaviours, with a transition between mean-field and low-dimensional regimes when $d=\min\{6,3α\}$, a transition between long- and short-range regimes at a crossover value $α_c(d)$, and with various logarithmic corrections at the boundaries between these regimes. This is the second of three papers developing a rigorous theory of the model's critical behavior in five of these eight regimes, including all long-range (LR) and high-dimensional (HD) regimes. We focus on the long-range low-dimensional (LR-LD) regime $d/3<α<α_c(d)$, where the model is below its upper critical dimension. Since computing $α_c(d)$ for $2<d<6$ appears to be beyond the scope of current techniques, we give an axiomatic definition of the LR regime which we prove holds for $α<1$. Using this, we prove up-to-constants estimates for the critical and slightly subcritical two-point function in the LR regime and for the volume tail and $k$-point function in the LR-LD regime. We deduce that the critical exponents satisfy the identities \[ η= 2-α, \qquad γ= (2-η)ν, \qquad \text{ and } \qquad Δ= νd_f \] in the LR regime (if $γ$, $ν$, or $Δ$ is well-defined) and that $δ$ and $d_f$ follow the hyperscaling identities \[ δ= \frac{d+α}{d-α} \qquad \text{ and } \qquad d_f = \frac{d+α}{2} \] in the LR-LD regime. Our results are suggestive of conformal invariance in the LR-LD regime, with the critical $k$-point function matching an explicit Möbius-covariant function up-to-constants.