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