Analysis of Hasse-Weil L-Function and Associated Entire Functions Applied in Mathematical Physics

In this paper, we study the analytic ranks of the Hasse–Weil [Formula: see text]-function. We consider the products for the completed Hasse–Weil [Formula: see text]-function in the framework of the theory of modular forms. We propose the new conjectures for the deformed Fourier sine and cosine integral formulas related to the completed Hasse–Weil [Formula: see text]-functions and Mittag-Leffler function. We show the entire-function solutions for the fractional diffusion equations in mathematical physics. We also discuss the trivial zeros of the Hasse–Weil [Formula: see text]-function.