  The spectral determinant is usually defined using the spectral zeta function
that is meromorphically continued to zero. In this definition, the complex
logarithms of the eigenvalues appear. Hence the notion of the spectral
determinant depends on the way how one chooses the branch cut in the definition
of the logarithm. We give results for the non-self-adjoint operators that state
when the determinant can and cannot be defined and how its value differs
depending on the choice of the branch cut.
