$$F(\eta)=-\frac{12 {\eta} \; _1F_2\left(\frac{1}{3};\frac{2}{3},\frac{4}{3};-\frac{{\eta}^3}{27}\right)}{6\Gamma \left(-\frac{1}{3}\right)}+\frac{\sqrt{3} {\eta}^2 \Gamma \left(\frac{2}{3}\right) \; _1F_2\left(\frac{2}{3};\frac{4}{3},\frac{5}{3};-\frac{{\eta}^3}{27}\right)}{6\pi }-\frac{1}{3}$$

$$F(\eta)\rightarrow 1+\sqrt[4]{3} \sqrt{2\pi}\frac{1}{x^{3/4}} \left[\sin \left(\frac{2 x^{3/2}}{3 \sqrt{3}}\right)-\cos \left(\frac{2 x^{3/2}}{3 \sqrt{3}}\right)\right],\;\;x\gg 1.$$