Existence of solution for nonlinear equation

Question

I would like to prove that the following equation, for a given $L>0$, $$\sinh \left(\sqrt{2} \sqrt{-\sqrt{1-8 a}-1} L\right) \sinh \left(\sqrt{2} \sqrt{\sqrt{1-8 a}-1} L\right)+\sqrt{-\sqrt{1-8 a}-1} \sqrt{\sqrt{1-8 a}-1} \cosh \left(\sqrt{2} \sqrt{-\sqrt{1-8 a}-1} L\right) \cosh \left(\sqrt{2} \sqrt{\sqrt{1-8 a}-1} L\right)-\sqrt{-\sqrt{1-8 a}-1} \sqrt{\sqrt{1-8 a}-1}=0$$ has only purely real or imaginary $a$ as solutions. Any suggestions to show the same would be helpful.