A product related to the cubic Gauss sum (Q1080456)

One of the many ways of deriving the law of quadratic reciprocity and determining the sign of the quadratic Gaussian sum rests on properties of the product \(\prod_{1\leq r\leq (p-1)/2}2 \sin (2\pi r/p)\). The author here describes an analogous cubic product and shows how it leads to the law of cubic reciprocity. He also restates his conjecture [Mat. Zametki 7, 469-476 (1970; Zbl 0194.350)] of a link between this product and the argument of the cubic Gauss sum.