Has anything similar to the Riemann hypothesis ever solved

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Has anything similar to the Riemann hypothesis ever been solved?

Specifically, has anyone proven that the real part of a result of some particular function always assumes a particular value?
 

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Specifically, has anyone proven that the real part of a result of some particular function always assumes a particular value?
It is not the function that takes particular values, the (nontrivial) zeros of the function (probably) occur at particular values.

It is easy to find some other examples, e.g. the zeros of the sine and cosine function are all on the real axis, the zeros of f(x)=ex-1 are all on the imaginary axis, and so on.
I don't know if there has been a function where such a proof has been found after considerable effort, but I would expect that there are hard examples.
 
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