We consider the inertial range spectrum of capillary wave turbulence. Under the assumptions of weak turbulence, the theoretical surface elevation spectrum scales with wave number k as I[subscript η] ∼ k[superscript α], where α = α[subscript 0] = -19/4, energy (density) flux P as P[superscript 1/2]. The proportional factor C, known as the Kolmogorov constant, has a theoretical value of C = C[subscript 0] = 9.85 (we show that this value holds only after a formulation in the original derivation is corrected). The k[superscript -19/4] scaling has been extensively, but not conclusively, tested; the P[superscript 1/2] scaling has been investigated experimentally, but until recently remains controversial, while direct confirmation of the value of C[subscript 0] remains elusive. We conduct a direct numerical investigation implementing the primitive Euler equations. For sufficiently high nonlinearity, the theoretical k[superscript -19/4] and P[superscript 1/2] scalings as well as value of C[subscript 0] are well recovered by our numerical results. For a given number of numerical modes N, as nonlinearity decreases, the long-time spectra deviate from theoretical predictions with respect to scaling with P, with calculated values of α C[subscript 0], all due to finite box effect
