  We investigate the scaling properties of the order parameter and the largest
nonvanishing Lyapunov exponent for the fully locked state in the Kuramoto model
with a finite number $N$ of oscillators. We show that, for any finite value of
$N$, both quantities scale as $(K-K_L)^{1/2}$ with the coupling strength $K$
sufficiently close to the locking threshold $K_L$. We confirm numerically these
predictions for oscillator frequencies evenly spaced in the interval $[-1, 1]$
and additionally find that the coupling range $\delta K$ over which this
scaling is valid shrinks like $\delta K \sim N^{-\alpha}$ with
$\alpha\approx1.5$ as $N \rightarrow \infty$. Away from this interval, the
order parameter exhibits the infinite-$N$ behavior $r-r_L \sim (K-K_L)^{2/3}$
proposed by Paz\'o [Phys. Rev. E 72, 046211 (2005)]. We argue that the
crossover between the two behaviors occurs because at the locking threshold,
the upper bound of the continuous part of the spectrum of the fully locked
state approaches zero as $N$ increases. Our results clarify the convergence to
the $N \rightarrow \infty$ limit in the Kuramoto model.
