$$ \frac{\zeta (0)}{\zeta (s)}= \prod _{p}\prod_{m= -\infty}^{\infty}\left(1-\frac{is\log p}{2\pi m}\right),$$
where $m$ does not run over $ m=0 $ and '$p$' means a product over all the primes :)
Is this valid ? I have used the Euler product representation
$$ \frac{1}{ \zeta (s)}= \prod _{p}(1-p^{-s}).$$
The zeros of $ 1-p^{-s} $ , are given by $ \frac{2\pi i m}{ \log(p)}$ so I think this would be valid , here $ i= \sqrt{-1} $.