Focusing on the issues of failing to pinpoint the edge information accurately and lacking texture detail of image by using integer order differential or 0-1-order fractional differential mask operators in digital image processing, a new 1-2-order edge detection operator based on Laplacian operator was proposed. Deduced from the definition of Riemann-Liouville (R-L),the 1-2-order fractional differential had the advantage in enhancing high-frequency signal and reinforcing medium frequency signal. The simulation results demonstrate that the proposed operator can take an higher recognition rate on the subjective recognition, and it's better at extracting the edge information, especially for the image with rich texture detail in the smooth region with little change of gray scale. Objectively, the integrated location error rate is 7.41% which is less than that of integer order differential operators (a minimum of 10.36%) and 0-1-order differential operator (a minimum of 9.97%). Quantitative indicators show the new fractional operator can effectively improve the positioning accuracy of the edge, and the proposed operator is particularly suitable for edge detection with high frequency information.