# zeta() Function & Examples

Use the zeta() special function in Calcul.io. Review its syntax, edit working examples, understand the result, and explore related math functions.

## zeta

#### zeta(s)

Try it yourself:

```calculio
zeta(0.2)
zeta(-0.5)
zeta(4)
```

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## Evaluate the Riemann zeta function

**zeta(s)** evaluates the Riemann zeta function, a special function related to the infinite series `1 + 1/2^s + 1/3^s + ...` where that series converges. Zeta appears in analytic number theory, probability, physics, and formulas involving sums of powers.

## Useful values and relationships

For example, `zeta(2)` equals `pi^2 / 6`. The function’s value at 1 is singular, so it is not an ordinary finite number. Use [pi](https://calcul.io/function/pi/index.md) to build well-known exact expressions, [pow](https://calcul.io/function/pow/index.md) for finite power calculations, and [gamma](https://calcul.io/function/gamma/index.md) when a related special-function formula calls for it.

## Series versus analytic continuation

The defining positive-term series converges only for real s greater than 1, but zeta is defined more broadly through analytic continuation. That distinction matters for negative and complex arguments: a calculator can return a valid continued value even though the simple series itself would not converge. Compare values numerically with [abs](https://calcul.io/function/abs/index.md) when working in the complex plane.

## Input caveats

Do not evaluate zeta at its pole s = 1. Values near that point can be extremely large and sensitive to precision. Zeta is a specialized mathematical function, not a generic way to sum a list; use [sum](https://calcul.io/function/sum/index.md) for finite data.

[All functions](https://calcul.io/functions/index.md)