Yes, but your LC circuit will need to be tuned to lock into the third overtone. It can be a sensitive setup which may loose lock under extreme situations like wide power supply voltage swings, or, extreme temperature change with cheap capacitors that drift in value with temperature.
Using an 74HC/74AC XOR gate, you can also do a 2x frequency doubling trick by feeding the output of any oscillator into input A, tying an inductor form input A to input B and add a varicap at the B input to GND to tune the B input to a 90 degree phase delay. The output of the XOR gate will be 2x the input frequency. With a 4x 2 input XOR gate chip, you can use 1 channel for the crystal oscillator, and another for the frequency doubling with 2 spare gates for buffers/inverters.