# z-function

`station__zeta__z-function` · external (needs EXECUTION_BACKEND_URL configured) · domain `zeta` · pv-relevance `non-pv`

Evaluate the Riemann-Siegel Z function Z(t), which is real-valued on the critical line. Sign changes of Z(t) indicate zeros of ζ(1/2 + it).

> **Note:** This tool routes through an external execution backend. If `EXECUTION_BACKEND_URL` is unset on the server, calls return JSON-RPC error `-32603 "Tool execution backend not configured"`. Tools with `backend: native` execute in-process and are always callable.

## Agent metadata

- `idempotent`: unknown
- `read_only`: unknown
- `expected_latency_ms`: unknown (not yet contract-tested)
- `cost_tokens_estimate`: unknown

## Input schema

- `t` *number* (required) — Height on the critical line

## Example call

```json
POST /api/mcp
Content-Type: application/json

{
  "jsonrpc": "2.0",
  "id": 1,
  "method": "tools/call",
  "params": {
    "name": "station__zeta__z-function",
    "arguments": {
      "t": 0
    }
  }
}
```

## Related

- [/tools](/tools) — all 3062 tools
- [/tools/zeta__z-function](/tools/zeta__z-function) — HTML page
- [/tools/zeta__z-function/json](/tools/zeta__z-function/json) — JSON form (agent-friendly)
- [/api/mcp](/api/mcp) — endpoint
- [/AGENTS.md](/AGENTS.md) — agent guide