Asymptotics for $|\alpha_k|$ is not sufficient to make conclusions about asymptotics of $f$. You need to know separately, the asymptotics on the positive and negative ray. Let us enumerate the roots $\alpha_k$ by all integers for convenience, and suppose that $\alpha_k\sim ak^{2/3},\; k\to +\infty$ and $\alpha_k\sim b(-k)^{2/3}, k\to-\infty$. Then there is a formula for asymptotics of $\log|f|$ of the form $$\log|f(x)|\sim h_1x^{3/2},\quad \log|f(-x)|\sim h_2|x|^{3/2},$$ as $ x\to+\infty$ outside a small exceptional set near the zeros.
Here $h_1$ and $h_2$ are constants which can be explicitly computed in terms of $a$ and $b$. See, for example, B. Levin, Distribution of zeros of entire functions, AMS, 1980, Chap. II. The condition $h_1<0,h_1<0$ is then sufficient for $f\in L^p$.
Actually it is easy to see that when both $a,b$ are positive, then both $h_j<0$, so $f\in L^p$ for all $p$.
The necessary conditions are of course $h_1\leq 0$ and $h_2\leq 0$. But when $h_j=0$ for some $j$, (or one of the $a,b$ $=0$), as is the case for the Airy function, the question becomes more subtle, and more precise asymptotics of zeros is needed to conclude $f\in L^p$.
If this is really what you need, I can compute $h_1$ and $h_2$ for you.
Remark. Your conditions imply that the Phragmen-Lindelöf indicator is $$h_f(\theta)=A\sin((3/2)(|\theta|-\alpha),\quad |\theta| \leq\pi,$$ where $A>0$ and $0\leq\alpha\leq \pi/3$. This makes my statement $h_j<0$ evident.
