  We study anomalous dimensions of unprotected low twist operators in the
four-dimensional $SU(N)$ $\mathcal{N}=4$ supersymmetric Yang-Mills theory. We
construct a class of interpolating functions to approximate the dimensions of
the leading twist operators for arbitrary gauge coupling $\tau$. The
interpolating functions are consistent with previous results on the
perturbation theory, holographic computation and full S-duality. We use our
interpolating functions to test a recent conjecture by the $\mathcal{N}=4$
superconformal bootstrap that upper bounds on the dimensions are saturated at
one of the duality-invariant points $\tau =i$ and $\tau =e^{i\pi /3}$. It turns
out that our interpolating functions have maximum at $\tau =e^{i\pi /3}$, which
are close to the conjectural values by the conformal bootstrap. In terms of the
interpolating functions, we draw the image of conformal manifold in the space
of the dimensions. We find that the image is almost a line despite the
conformal manifold is two-dimensional. We also construct interpolating
functions for the subleading twist operator and study level crossing phenomenon
between the leading and subleading twist operators. Finally we study the
dimension of the Konishi operator in the planar limit. We find that our
interpolating functions match with numerical result obtained by Thermodynamic
Bethe Ansatz very well. It turns out that analytic properties of the
interpolating functions reflect an expectation on a radius of convergence of
the perturbation theory.
