Resonance Frequency of system with two inductors
The impedance of the parallel RLC is known, and it is obtained by putting together the three admittances and then reciprocating:
z1 = 1/(1/R + jwC + 1/jwL2)
The you add the inductor L1 in series
z2 = z1 + jwL1
This is the total impedance.
Then the resonance frequencies correspond to the poles of the circuit (Note: if the poles are real, ok , no resonance; if they are complex, you consider the imaginary part). Now, it's easier to take the zeros of the admittance: y2 = 1/z2
These zeros are the indication of the resonance frequencies of the circuit.