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This paper proves that the imaginary part of the Riemann $\xi$ function isstrictly monotonic with $b$ in the region $S = \{t|t=a+bi,\ 0\leq a \leq9.508,\ -1/2<b<1/2\}$. That leads to Im($\xi$)=0 being true only when $b=0$ in$S$.

Monotonicity of the imaginary part of the Riemann $\xi$ function in the region $S$