  We reconsider black hole solutions of D = 3 higher-spin gravity in the
$hs[\lambda]\oplus hs[\lambda]$ Chern-Simons formulation. A suitable
generalisation of the BTZ black hole has a spin-3 chemical potential $\alpha$,
and non-zero values of all the conserved charges associated with the asymptotic
$\mathcal W_\infty[\lambda]$ symmetry. We extend the available perturbative
expansion of the partition function to order $\mathcal O(\alpha^{18})$ for
generic values of the $\lambda$ parameter. The result matches the CFT
prediction at $\lambda=0$ and at $\lambda=1$ where we provide the exact
all-order expansion of the partition function. The perturbative series is then
analysed in the interesting non-trivial limit $\lambda\to\infty$ and we derive
the exact analytical expressions of the partition function and the spin-4
charge in closed form as functions of $\alpha$. Also, the first subleading
correction at large $\lambda$ is shown to be simply related to the leading
contribution.
