Angular momentum (L) in z-axis of hydroxyl radical

[Nezva]

Problem:
A hydroxyl radical in the gas phase is found to be in a rotational state with the wavefunction given, it's a long wavefunction so I'm not going to post unless needed.

a) Calculate the expected value of L_z for this radical in SI units.
I think L_z=m_lħ is important for this but where does the given psi function come into play? I doubt it is extraneous information.

Information:
ħ = h/2$$\pi$$
$$h\ =\ 6.62606876(52)\ \times\ 10^{-34}\ J\ s$$
m_l = -l, -l+1, -l+2, ... l-2, l-1, l
My problem: Unsure how to find the proper orbitals associated with OH radical. The hydrogen is 1s¹ and the oxygen is 1s² 2s² 2p⁴.

b) Is the angular momentum vector parallel, perpendicular, or at an angle from the z-axis.

I'll attempt a solution if someone can point me in the right direction, thank you.

 

[Nezva]

I'm still struggling on this problem.

How does the lone electron effect the orbitals associated with the molecule?
(L_z)

Beginning attempt at solution: (possibly incorrect formulas)

The orbitals would give me the rotational quantum number (l), thus giving me spatial orientation quantum number (m_l), with which I could solve for L_z=m_lħ