Help me find the zeros of f(z)

Let's define f(z) over the complex plane where 0 < a < 1.

Such that z= a +b*i, where a and b are real numbers, and i is the imaginary number i = sqrt(-1). (z is complex by definition, unless b = 0 then it's real)

Where also known as the Gamma Function over complex numbers.

The question is for what values is f(z) = 0?

N.B: Zeros have been found for a= 1/2. But for no other 'a', nonetheless zeros might exist elsewhere, help me find all the zeros.

Thanks :)