In the present paper, we construct theta functions with two parameters a, b R that satisfy Jacobi's modular relation. Furthermore, we define zeta functions, also depending on two real parameters a, b R and which are derived from these theta functions, which satisfy Riemann's functional equation. To the best of our knowledge, these zeta functions are the first known examples that simultaneously satisfy Riemann's functional equation and involve two independent parameters.
T. Nakamura (Thu,) studied this question.
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