# Closed Forms of Certain Zeta constants?

The Riemann Zeta function $$\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}$$ converges for $$\operatorname{Re}(s)>1$$. I am interested in some particular "irrational " values of it such as:

• $$\zeta(\pi)=1.176241738\ldots$$,
• $$\zeta(e)=1.2690096043\ldots$$,
• $$\zeta(\sqrt2)=3.020737679\ldots$$,
• $$\ldots$$

Are there closed form representations for these and constants? Are there formulas which consists of these constants?

• The Riemann Zeta Function exists for all complex numbers except for $\;z=1\;$ ....Perhaps you meant the summatory form $$\sum_{n=1}^\infty\frac1{n^s}\;,\;\;\text{which converges for Re}\,(s)>1\; ?$$ – DonAntonio Sep 28 '13 at 16:32
• Yes that. Specially !and @DonAntonio does the Riemann zeta function exists for $0$ too? – Shivam Patel Sep 28 '13 at 16:36
• Yes Shivam: for any complex value except $\;1\;$ . – DonAntonio Sep 28 '13 at 16:42
• In some sense, $\zeta(\sqrt{2})$ is already a closed form. – Hurkyl Sep 28 '13 at 17:32

## 1 Answer

There is no reason to suspect that these have a "closed form". There isn't even a known closed form for $\zeta(3)$...