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Wikipedia states that

The Riemann zeta function $\zeta(s)$ is defined for all complex numbers $s \neq 1$. It has zeros at the negative even integers (i.e. at $s = −2, −4, −6, ...)$. These are called the trivial zeros. The Riemann hypothesis is concerned with the non-trivial zeros, and states that: The real part of any non-trivial zero of the Riemann zeta function is $\frac{1}{2}$.

What does it mean to say that $\zeta(s)$ has a $\text{trivial}$ zero and a $\text{non-trivial}$ zero. I know that $$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$$ what wikipedia claims it that $\zeta(-2) = \sum_{n=1}^{\infty} n^{2} = 0$ which looks absurd.

My question is can somebody show me how to calculate a zero for the $\zeta$ function.

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    I think it may help http://mathworld.wolfram.com/Riemann-SiegelFormula.html2012-04-20
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    The series is not applicable for $\Re(s)\leq 1$; one uses a different formula (an *analytic continuation*, if you will) of the $\zeta$ function (so yes, it does look absurd until you consider the extension of the function to the rest of the complex plane).2012-04-20
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    As Ginger mentions, one uses the Riemann-Siegel formula *numerically* to compute the nontrivial zeroes (there are no known closed forms for them).2012-04-20

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