\[\exp\left [ 2\psi(2\pi)-\psi(\pi)-\psi \left (\pi+\dfrac12 \right)\right ] = \, ? \]

**Hint**: Legendre duplication formula.

**Notations**:

\( \exp[x] \) denotes the exponential function, \( \exp[x] = e^x \).

\(\psi(\cdot)\) denotes the digamma function.

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