= Solution
The nonzero poles are
$$
t=\pm ia_n,\qquad a_n=\frac\pi2(2n+1),\qquad n\geq0.
$$
Their paired residues sum to
$$
-(-1)^na_n^{s-1}\cos\frac{\pi s}{2}.
$$
The paired <series> converges for $\operatorname{Re}s<1$ by the alternating-series test, and absolutely for $\operatorname{Re}s<0$. Therefore
$$
\sum\operatorname{Res}=-\cos\frac{\pi s}{2}
\left(\frac\pi2\right)^{s-1}\beta(1-s).
$$
Using the stated contour closure, the Hankel <integral> is $-2\pi i$ times this sum. Substitution into part (a) gives
$$
\beta(s)=\Gamma(1-s)\cos\frac{\pi s}{2}
\left(\frac\pi2\right)^{s-1}\beta(1-s).
$$
Now $1/\Gamma(1-s)=\Gamma(s)\sin(\pi s)/\pi$ and $\sin(\pi s)=2\sin(\pi s/2)\cos(\pi s/2)$, so
$$
\boxed{\beta(1-s)=\Gamma(s)\left(\frac\pi2\right)^{-s}
\sin\frac{\pi s}{2}\,\beta(s)}.
$$
It was derived on a nonempty half-plane. Both sides have analytic continuations, with the apparent gamma poles cancelled by the sine factor or the trivial zeros from part (b); the <identity theorem> therefore extends it to every $s\in\mathbb C$.
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