  This short note for non-experts means to demystify the tasks of evaluating
the Riemann Zeta Function at non-positive integers and at even natural numbers,
both initially performed by Leonhard Euler. Treading in the footsteps of G. H.
Hardy and others, I re-examine Euler's work on the functional equation for the
Zeta function, and explain how both the functional equation and all `classical'
integer values can be obtained in one sweep using only Euler's favorite method
of generating functions. As a counter-point, I also present an even simpler
argument essentially due to Bernhard Riemann, which however requires Cauchy's
residue theorem, a result not yet available to Euler. As a final point, I
endeavor to clarify how these two methods are organically linked and can be
taught as an intuitive gateway into the world of Zeta functionology.
