Integrating $e^{-(t|k|^\alpha+ixk)} $ with respect to $k$

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Is there any way to calculate below integral ?: $$ \int_{-\infty}^{ \infty} \exp\left(-t\left\vert\, k\,\right\vert^{\alpha} - \,\mathrm{i}kx\right) \,\mathrm{d}k\qquad\mbox{where}\quad 0 < \alpha < 2 $$