About the adjoint of some differential operator on $L^2(0,1)$

Question

If $T=-d^2/dx^2$ is defined on the domain $D(T)=\{f\in C^2[0,1]: f(1)=f'(0)=f'(1/2)=0\}\subset L^2(0,1)$. What's the Hilbert adjoint operator $T^*$?

Many many thanks for your answers.

Math.