Is this divergent infinite product meaningful?

Using this assumption, which comes as an intuition from working with digital audio:

• The value (magnitude and phase) of a complex-domain function at any point $z$, is directly proportional to the differences $z-Z_i$ for all the roots, and inversely proportional to the differences $z-P_i$ for all the poles of the function.
  • This is certainly true for any algebraic function, and I believe it is also true for well-behaved transcendental functions.

And these received facts:

• The trivial roots of the Riemann zeta function are at $-2,-4,-6\dots$ .
• The non-trivial roots are symmetric around the line with real part $\frac{1}{2}$ .
• The only pole of the zeta function is at 1.
• $\zeta (2)=\frac{{\pi }^2}{6}$ .
• $\zeta (-1)=-\frac{1}{12}$ .

Produce these relations: $$\frac{{\pi }^2}{6}=\zeta (2)=\frac{C}{1}\times 4\times 6\times 8\times \dots$$
$$-\frac{1}{12}=\zeta (-1)=-\frac{C}{2}\times 1\times 3\times 5\times \dots$$

The unknown (notionally infinite!) constant $C$ resulting from the non-trivial zeros can be eliminated by combining the equations above:

$$\frac{\zeta (2)}{\zeta (-1)}=-2{\pi }^2=-\frac{2}{1}\times \frac{4\times 6\times 8\times \dots }{1\times 3\times 5\times \dots }$$

This looks similar to the Wallis product, but it is badly divergent. Does this formal product have any interpretation?