In $\zeta (s) $, $s = \sigma + ti $, where $t $ is a real number, what would become of the zeta function if $t $ were to be complex?
Does there exist a proof that $\zeta (\sigma + (a+bi)i) $ cannot be zero, where $(a+bi) = t $, $a $ is real, and b is either real or complex?
Does $t $ have to be a real number for $\zeta (\sigma+ti) $ to vanish? Please provide a reference for the proof of this.