$\textbf{Answer}$

The answer is no.

$\textbf{Analysis}$

Let's color the board in the following way:

We note that regardless of how a L-shaped piece is placed over this board, it will cover $1$ square of one color and $3$ squares of the other color. In another word, placing one L-shaped piece will result in a difference of $2$ in the number of different colored squares covered. Because $15$ is an odd number, therefore placing $15$ such pieces cannot make the difference become zero. However, this board has the same number of black and white squares. Hence, the answer is no.

$\textbf{Note}$

This problem is of the same nature as # 2744 but is more challenging. It also shows that the key to employ the coloring method is to design an appropriate coloring scheme.