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This paper treats about one of the most remarkable achievements by Riemann,that is the symmetric form of the functional equation for {\zeta}(s). Wepresent here, after showing the first proof of Riemann, a new, simple anddirect proof of the symmetric form of the functional equation for both theEulerian Zeta function and the alternating Zeta function, connected with oddnumbers. A proof that Euler himself could have arranged with a little step atthe end of his paper "Remarques sur un beau rapport entre les s\'eries despuissances tant direct que r\'eciproches". This more general functionalequation gives origin to a special function, here named {\cyr \E}(s), which weprove that it can be continued analytically to an entire function over thewhole complex plane using techniques similar to those of the second proof ofRiemann. Moreover we are able to obtain a connection between Jacobi's imaginarytransformation and an infinite series identity of Ramanujan. Finally, afterstudying the analytical properties of the function {\cyr \E}(s), we completeand extend the proof of a Fundamental Theorem, both on the zeros of RiemannZeta function and on the zeros of Dirichlet Beta function, using also theEuler-Boole summation formula.

An Alternative Form of the Functional Equation for Riemann's Zeta Function, II