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I'd like to introduce a function, which I call Y function, just because that's how its plot look like in complex plane. It can also called upsilon function, just to sound more geeky. It's defined as follow:Y(s)=ln{ζ(s) - ζ(1-s*)}
When m =1, the low slope part isn't visible.
I played around with this Y function, and found something interesting when 1 in the formula is replaced with other numbers. The plot looks simpler when s has high imaginary part.
The logarithm of a complex number z=log(r*e^(iθ)) is condensed as log(z)=log(r)+iθ, where r is the absolute value of z and θ is the argument of z. It is represented in polar coordinates.
when imaginary part of s is close to the smallest non-trivial zero of Riemann's zeta functionhttps://www.wolframalpha.com/input?i=plot+%28ln%28%28zeta%28s%2B14.134725+i%29%29+-+%28Zeta%281-s%2B14.134725+i%29%29%29+%29from+-22+to+23Any larger value for the imaginary part of s don't change shape of the real part of Y function, only the imaginary part gets longer period.https://www.wolframalpha.com/input?i=plot+%28ln%28%28zeta%28s%2B114.134725+i%29%29+-+%28Zeta%281-s%2B114.134725+i%29%29%29+%29from+-22+to+23