Abstract: This paper is devoted to the problem of constructing L-optimal designs for a trigonometric Fourier regression model with no intercept. The paper considers diagonal matrices L with a combination of zeros and ones on the main diagonal. It is shown that in the case when L = I (i.e., when the identity matrix is chosen as the matrix L), the L-optimal design coincides with the D-optimal one. In the more general case (when some diagonal elements are zero), it is shown that the dimension of the problem can be reduced if the optimal design is symmetric. The results are illustrated by the example of the problem of constructing two L-optimal designs for the twelfth-order trigonometric model, which is reduced to the problem of constructing designs for third- and fourth-order models, correspondingly.