Landau-Siegel Zeros of Triple Product L-functions

Let $F$ be a number field. Let $\pi_1,\pi_2$ be cuspidal automorphic representations of $GL_2(\mathbb{A}_F)$, and let $\pi$ be a cuspidal automorphic representation of either $GL_2(\mathbb{A}_F)$ or $GL_3(\mathbb{A}_F)$. When $(\pi_1,\pi_2,\pi)$ is of general type, we show that the triple product $L$-function $L(s,\pi_1 \times \pi_2 \times \pi)$ on either $GL(2) \times GL(2) \times GL(2)$ or $GL(2) \times GL(2) \times GL(3)$ has a standard zero-free region with no exceptional Landau-Siegel zero. Moreover, when $(\pi_1,\pi_2,\pi)$ is not of general type, we give precise conditions when $L(s,\pi_1 \times \pi_2 \times \pi)$ could possibly have exceptional Landau-Siegel zeros.