In the course of studying a certain complex-valued functional equation, I have had a need to evaluate the following limit:

$$\gamma_\mathcal{T}=\lim_{n\to\infty}\left(-\frac{i}{2}\sum_{k=1}^n \frac1{ik+k^{3/2}}-\log\left(1+\frac{i}{\sqrt n}\right)\right)$$

which is structurally similar to the usual limit definition for the Euler constant $\gamma$.

So far as I can tell, there seems to be no elementary closed form for this limit, so I set about trying for numerical estimation.

The problem is that the convergence of this limit looks to be excruciatingly slow. Even with the help of a sequence extrapolation method, I only managed to produce a few good digits:

$$\gamma_\mathcal{T}\approx-0.5-0.9300125396i$$

I am wondering if there are more efficient, alternative methods for numerically evaluating this limit. Thanks in advance!