L-functions with Riemann's functional equation and the Riemann hypothesis.

The article focuses on constructing and analyzing new L-functions, denoted \( R_j(s) \) for \( j=1,2,3 \), which satisfy both Riemann's functional equation and an analogue of the Riemann hypothesis. These functions are defined using non-principal Dirichlet characters modulo 3 and 4 and are shown to have zeros only at non-positive even integers and complex numbers with real part \( 1/2 \). The paper proves that each \( R_j(s) \) fulfills the functional equation \( R_j(1-s) = \Gamma_{\cos}(s) R_j(s) \), where \( \Gamma_{\cos}(s) \) is the gamma factor appearing in the classical Riemann zeta function's functional equation, and establishes zero-free regions for \( \Re(s) > 1/2 \). Additionally, the article provides infinite product representations and defines analogues of Hardy's Z-function for these \( R_j(s) \), supported by numerical illustrations. This work contributes explicit examples of L-functions that simultaneously satisfy Riemann's functional equation and the Riemann hypothesis, a combination not previously achieved.