= Solution
At a nonpositive integer, the branch disappears and the collapsed Hankel contour extracts the residue at zero. For $s=0$,
$$
\beta(0)=\operatorname{Res}_{t=0}\frac1{t(e^t+e^{-t})}=\frac12.
$$
For $s=-2$, the coefficient of $t^{-1}$ in
$$
t^{-3}\left(\frac12-\frac14t^2+O(t^4)\right)
$$
is $-1/4$; multiplication by $\Gamma(3)=2$ gives
$$
\beta(-2)=-\frac12.
$$
When $s=-2n-1$, the integrand is $t^{-2n-2}$ times an even power <series>, so it has no $t^{-1}$ coefficient. Hence
$$
\boxed{\beta(-2n-1)=0\qquad(n\geq0).}
$$
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