In a solenoid, if number of turns per unit length is doubled, then self inductance will become:

half of its initial value double of its initial value $\frac{1}{4}$ times of its initial value 4 times of its initial value

4 times of its initial value

Coefficient of Self inductance is directly proportional to square of number of turns in coil, hence on doubling the number of turns. Coefficient of Self inductance becomes four times.