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I know values of zeta for s= 2,3,4,... but what's the value of zeta as an example s= 2+14i

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    Yes the series $\sum_{n=1}^\infty n^{-s}$ converges whenever $\mathrm{Re}(s)>1$, regardless of the imaginary part of $s$. A standard comparison test should suffice. Also, the body of your question seems to be asking for explicit values of the zeta function at nonreal arguments. Do you want numerical values (which e.g. WolframAlpha can spit out easily), or closed-form symbolic expressions? (If the latter, I have a hard time believing you know the value of $\zeta$ for $s=3$.)2012-10-17
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    Probably Mr Chuy should spend some more time working on analysis ... it will surely help him if he wants to work on the zeta function.2012-11-08

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