I have performed some computations in wolfram alpha for $z=\log (cosx)+i\log (sinx) $ at $x = 1.552 \cdots$ I got this result , Really i don't have enough competence to know what is the real diffrence between Non zero and trivial zero because as show above in the titled question $z=-3.99\cdots+0.0001\cdots $ has a real part's almost is trivial zero which is :the integer $x=-4$.
My question here :Can I say that :$\zeta(-3.9901894525-i 0.0001710810718$ ) is a non zero ?
Note: At a least I would like to know how do i can considerating it
Thank you for any help !!!!
I recommend you go back to the paper in which the Riemann zeta function was introduced.
http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZeta.pdf
Right off the bat (first paragraph of page 4 by footer numbers - NB the definition of ξ a few lines before), it was established that any nontrivial zeroes must have a real part between 0 and 1. If you're going to be tackling the most notorious open problem in mathematics, you should probably endeavor in humility to read and understand not only that paper, but reams and reams of further material on the subject.