# Are there known zeros of the Zeta function off the line 1/2?

I've been doing research in information theory that had some superficial connections to the Zeta function, since I was making use of logarithms of complex numbers.

In short, we'd be looking at the Zeta function using values of s produced by powers of the logarithm function:

$$\zeta(\log^n (-1)/x).$$

I noticed several zeros for n = 6:

https://www.wolframalpha.com/input/?i=plot+zeta(log%5E6(-1)%2Fx),+x+from+1+to+500

But, I don't know enough about this topic to be sure that this is actually interesting, since I know that at least some of the zeros of this function are considered "trivial". Nonetheless, I thought I'd share this in case some of you found it interesting and had insights.

• $\zeta(-2n)=0$ are "trivial" zeros off the line $1/2$, see this post. – Dietrich Burde Dec 28 '18 at 15:42
• There are no known zeros of $\zeta(s)$ that don’t occur at a negative even integer (these are the trivial zeros) or have real part $\sigma=\frac12$. The Riemann Hypothesis states that any zero with real part between $0$ and $1$ will in fact have real part exactly $\frac12$. – Clayton Dec 28 '18 at 15:42
• For more on the zeroes and the history of zero finding of the Riemann zeta function, I once geeked out and wrote a fair synopsis. – Eric Towers Dec 28 '18 at 19:17

The only known zeroes of the zeta function off of the $${1 \over 2} + a\times i$$ line are the negative even integers, each of which is a zero. These are called trivial zeroes. The Riemann Hypothesis states that all other zeroes lie on that magic line. It is known that at least 40% of them lie there, and none off of it have ever been found.