Consider a number $z_0\in\mathbb{C}$ such that
$$<br /> z_0=-(\aleph_\mathcal{X}+b)+iy~~,\quad\text{where}\qquad b,y\in\mathbb{R}_0~~.<br />$$

Observe that the Euler product form of $\zeta$takes $z_0$ as
$$<br /> \zeta(z_0)=\prod_{p}\cfrac{1}{1-p^{(\aleph_\mathcal{X}+b)-iy}}\nonumber\\<br /> \qquad= \left(\cfrac{1}{1-P^{(\mathcal{X}\aleph_1+b)-iy}} \right) \prod_{p\neq P}\cfrac{1}{1-p^{(\aleph_\mathcal{X}+b)-iy}} \nonumber\\<br /> \qquad=\left(\cfrac{1}{1-\cfrac{\vphantom{\hat1}1}{P^b}\big(P^{\mathcal{X}}\big)^{\!\widehat\infty}\left[ \cos(y\ln P)-i\sin(y\ln P)\right]} \right) \prod_{p\neq P}\cfrac{1}{1-p^{(\aleph_\mathcal{X}+b)-iy}} \nonumber~~.<br />$$

Let $y\ln P=2n\pi$ for some prime $P$ and $n\in\mathbb{N}$ or $n=0$. Then
$$<br /> \zeta(z_0)=\cfrac{1}{\infty} \prod_{p\neq P}\cfrac{1}{1-p^{(\aleph_\mathcal{X}+b)-iy}}=0~~. <br />$$