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We compute the one-loop free energies of the type-A$_\ell$ and type-B$_\ell$higher-spin gravities in $(d+1)$-dimensional anti-de Sitter (AdS$_{d+1}$)spacetime. For large $d$ and $\ell$, these theories have a complicated fieldcontent, and hence it is difficult to compute their zeta functions using theusual methods. Applying the character integral representation of zeta functiondeveloped in the companion paper arXiv:1805.05646 to these theories, we showhow the computation of their zeta function can be shortened considerably. Wefind that the results previously obtained for the massless theories ($\ell=1$)generalize to their partially-massless counterparts (arbitrary $\ell$) inarbitrary dimensions.

Character Integral Representation of Zeta function in AdS$_{d+1}$: II. Application to partially-massless higher-spin gravities