Can someone help me find the zeros of this function?

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adr58_1 2019-10-19

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/486339-can-someone-help-me-find-the-zeros-of-this-function

Let's define f(z) over the complex plane except where a =1. N.B.: z= a +b*i, where a and b are real numbers, and i is the imaginary number i = sqrt(-1).

f(z) = {\displaystyle \f (s)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {x^{s-1}}{e^{x}-1}}\,\mathrm {d} x}

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

The values where z= -2*k + b*i, where k is a natural integer have been shown to be zeros for every b element of real numbers,

but there are other zeros such that if complex 'c' is a zero of f(z) then 'c = q + b*i', where q is a real number such as '0 < q < 1' and b is an unrestricted real number.

Help me find all the zeros please,

Thanks