# Riemann Zeta Function at Even Integers/Examples

## Examples of Riemann Zeta Function at Even Integers

### Riemann Zeta Function of $2$

The Riemann zeta function of $2$ is given by:

 $\ds \map \zeta 2$ $=$ $\ds \dfrac 1 {1^2} + \dfrac 1 {2^2} + \dfrac 1 {3^2} + \dfrac 1 {4^2} + \cdots$ $\ds$ $=$ $\ds \dfrac {\pi^2} 6$ $\ds$ $\approx$ $\ds 1 \cdotp 64493 \, 4066 \ldots$

### Riemann Zeta Function of $4$

The Riemann zeta function of $4$ is given by:

 $\ds \map \zeta 4$ $=$ $\ds \dfrac 1 {1^4} + \dfrac 1 {2^4} + \dfrac 1 {3^4} + \dfrac 1 {4^4} + \cdots$ $\ds$ $=$ $\ds \dfrac {\pi^4} {90}$ $\ds$ $\approx$ $\ds 1 \cdotp 08232 \, 3 \ldots$

### Riemann Zeta Function of $6$

The Riemann zeta function of $6$ is given by:

 $\ds \map \zeta 6$ $=$ $\ds \dfrac 1 {1^6} + \dfrac 1 {2^6} + \dfrac 1 {3^6} + \dfrac 1 {4^6} + \cdots$ $\ds$ $=$ $\ds \dfrac {\pi^6} {945}$ $\ds$ $\approx$ $\ds 1 \cdotp 01734 \, 3 \ldots$

### Riemann Zeta Function of $8$

The Riemann zeta function of $8$ is given by:

 $\ds \map \zeta 8$ $=$ $\ds \dfrac 1 {1^8} + \dfrac 1 {2^8} + \dfrac 1 {3^8} + \dfrac 1 {4^8} + \cdots$ $\ds$ $=$ $\ds \dfrac {\pi^8} {9450}$ $\ds$ $\approx$ $\ds 1 \cdotp 00408 \, 3 \ldots$

### Riemann Zeta Function of $26$

The Riemann zeta function of $26$ is given by:

 $\ds \map \zeta {26}$ $=$ $\ds \dfrac 1 {1^{26} } + \dfrac 1 {2^{26} } + \dfrac 1 {3^{26} } + \dfrac 1 {4^{26} } + \cdots$ $\ds$ $=$ $\ds \dfrac {\pi^{26} \times 2^{24} \times 76 \, 977 \, 927} {27!}$