# Patterns in zeta function

1. May 13, 2006

### heartless

I'm experimenting with zeta function right now, and I assume there must be some kind of patter in zetas of consecutive (even) numbers.

For example when we do,
$$\zeta(2)=\pi^2 /6$$

$$\zeta(4)=\pi^4/90$$

$$\zeta(6)=\pi^6/945$$

$$\zeta(8)=\pi^8/9450$$

However,

$$\zeta(12)=691\pi^{12}/638512875$$

So, Can anyone explain this pattern to me?
I'd really appreciate

Last edited: May 13, 2006
2. May 13, 2006

### Curious3141

3. May 14, 2006

### heartless

Thanks Curious,

4. May 14, 2006

### benorin

$$\zeta (2m) = (-1)^{n+1}\frac{(2\pi )^{2m}B_{2m}}{2(2m)!}$$

where $$B_n$$ is the nth Bernoulli number.