  Consider the long-range percolation model on the integer lattice
$\mathbb{Z}^d$ in which all nearest-neighbour edges are present and otherwise
$x$ and $y$ are connected with probability $q_{x,y}:=1-\exp(-|x-y|^{-s})$,
independently of the state of other edges. Throughout the regime where the
model yields a locally-finite graph, (i.e.\ for $s>d$,) we determine the
spectral dimension of the associated simple random walk, apart from at the
exceptional value $d=1$, $s=2$, where the spectral dimension is discontinuous.
Towards this end, we present various on-diagonal heat kernel bounds, a number
of which are new. In particular, the lower bounds are derived through the
application of a general technique that utilises the translation invariance of
the model. We highlight that, applying this general technique, we are able to
partially extend our main result beyond the nearest-neighbour setting, and
establish lower heat kernel bounds over the range of parameters $s\in (d,2d)$.
We further note that our approach is applicable to short-range models as well.
