It is now believed that the scaling exponents of moments of velocity increments are anomalous, or that the departures from Kolmogorov's (1941) self-similar scaling increase nonlinearly with the increasing order of the moment. This appears to be true whether one considers velocity increments themselves or their absolute values. However, moments of order lower than 2 of the absolute values of velocity increments have not been investigated thoroughly for anomaly. Here, we discuss the importance of the scaling of non-integer moments of order between +2 and $-1$, and obtain them from direct numerical simulations at moderate Taylor microscale Reynolds numbers $R_\lambda\le$ 450, and experimental data at high Reynolds numbers $(R_\lambda \approx 10\,000)$. The relative difference between the measured exponents and Kolmogorov's prediction increases as the moment order decreases towards $-1$, thus showing that the anomaly is manifested in low-order moments as well.