  We investigate the 3-sphere partition functions of various 3d $\mathcal{N}=2$
holographic SCFTs arising from the $N$ stack of M2-branes in the 't Hooft limit
both analytically and numerically. We first employ a saddle point approximation
to evaluate the free energy $F=-\log Z$ at the planar level, tracking the first
subleading corrections in the large 't Hooft coupling $\lambda$ expansion.
Subsequently, we improve these results by determining the planar free energy to
all orders in the large $\lambda$ expansion via numerical analysis. Remarkably,
the resulting planar free energies turn out to take a universal form,
supporting a prediction that these $S^3$ partition functions are all given in
terms of an Airy function even beyond the special cases where the Airy formulae
were derived analytically in the literature; in this context we also present
new Airy conjectures in several examples. The subleading behaviors we captured
encode a part of quantum corrections to the M-theory path integrals around dual
asymptotically Euclidean AdS$_4$ backgrounds with the corresponding internal
manifolds through holographic duality.
