Regulator of the Hesse cubic curves and hypergeometric functions

We construct some integral elements in the motivic cohomology of the Hesse cubic curves and express their regulators in terms of generalized hypergeometric functions and Kamp\'e de F\'eriet hypergeometric functions. By using these hypergeometric expressions, we obtain numerical examples of the Bloch-Beilinson conjecture on special values of $L$-functions.