  The Quantum Spectral Curve (QSC) equations for planar $\mathcal{N}=6$
super-conformal Chern-Simons (SCS) are solved numerically at finite values of
the coupling constant for states in the $\mathfrak{sl}(2|1)$ sector. New weak
coupling results for conformal dimensions of operators outside the
$\mathfrak{sl}(2)$-like sector are obtained by adapting a recently proposed
algorithm for the QSC perturbative solution. Besides being interesting in their
own right, these perturbative results are necessary initial inputs for the
numerical algorithm to converge on the correct solution. The non-perturbative
numerical outcomes nicely interpolate between the weak coupling and the known
semiclassical expansions, and novel strong coupling exact results are deduced
from the numerics. Finally, the existence of contour crossing singularities in
the TBA equations for the operator $\textbf{20}$ is ruled out by our analysis.
The results of this paper are an important test of the QSC formalism for this
model, open the way to new quantitative studies and provide further evidence in
favour of the conjectured weak/strong coupling duality between $\mathcal{N}=6$
SCS and type IIA superstring theory on $AdS_4 \times CP^3$. Attached to the
arXiv submission, a Mathematica implementation of the numerical method and
ancillary files containing the numerical results are provided.
