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I am interested in what happens to the zeta function zeroes if the real part is made slightly greater than or less than $1/2$.

What I am asking is: can someone graph a few nonzero values of $\zeta(s)$?

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    See http://www.wolframalpha.com/input/?i=Plot%5B%7BRe%5BZeta%5Bt%2BZetaZero%5B1%5D-1%2F2%5D%5D,+Im%5BZeta%5Bt%2BZetaZero%5B1%5D-1%2F2%5D%5D%7D++,++%7Bt,0,1%7D%5D – 2017-02-02

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Pictures can be found on the Wikipedia article on the zeta function https://en.wikipedia.org/wiki/Riemann_zeta_function