What is the pattern in the zeta function for consecutive even numbers?

[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

 

[Curious3141]

You can see the pattern from this link : http://functions.wolfram.com/10.01.03.0003.01

and for the definition of the Bernoulli Numbers : http://mathworld.wolfram.com/BernoulliNumber.html

 

[heartless]

Thanks Curious,

 

[benorin]

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

where $$B_n$$ is the n^th Bernoulli number.