When Does \(\displaystyle\int_{-1}^0\text{sexp}(x)\,\mathrm{d}x\) Equal \(\frac12\)?

[Catullus]

With the linear approximation of tetration, \(\displaystyle\int_{-1}^0\text{sexp}(x)\,\mathrm{d}x=\frac12\).
For what complex base(s) does \(\displaystyle\int_{-1}^0\text{sexp}(x)\,\mathrm{d}x\) equal \(\frac12\) with the the analytic continuation of the Kneser method?
Also, for what complex base(s) does \(\text{sexp}(-\frac12)\) equal \(\frac12\) with the same method of tetration?