This is not a question but I hope and wish that it will not be deleted too fast.
Today, I received an upvote for my answer to this question. Not remembering what the problem was (age, for sure !), I had a look at it and I was surprised to see that the superb approximation @robjohn made answering this question of mine (four years ago) was giving a rather poor estimate of one of the solutions of equation $\color{red}{\Gamma(z)=i}$.
Redoing my calculations, I found a terrible numerical mistake.
As a tribute to @robjohn (hoping that he will forgive me), I report here the solution obtained for $$\color{red}{\Gamma(z)=n\,i}$$ for which the estimate is given by $$\color{blue}{z\sim e^{1+W(t)}+\frac 12}\qquad \text{where} \qquad \color{blue}{t=\frac 1 e \log \left(\frac{n\,i}{\sqrt{2 \pi }}\right)}$$ $W(t)$ being Lambert function.
Some impressive results
| $n$ | Approximation | Solution |
|---|---|---|
| $1$ | $2.84071 +1.69496 \,i$ | $2.84550 +1.68429 \,i$ |
| $2$ | $3.34357 +1.44741 \,i$ | $3.35044 +1.43963 \,i$ |
| $3$ | $3.64315 +1.33808 \,i$ | $3.65025 +1.33195 \,i$ |
| $4$ | $3.85317 +1.27397 \,i$ | $3.86019 +1.26882 \,i$ |
| $5$ | $4.01366 +1.23065 \,i$ | $4.02051 +1.22614 \,i$ |
| $6$ | $4.14297 +1.19882 \,i$ | $4.14966 +1.19475 \,i$ |
| $7$ | $4.25095 +1.17409 \,i$ | $4.25749 +1.17037 \,i$ |
| $8$ | $4.34346 +1.15414 \,i$ | $4.34987 +1.15068 \,i$ |
| $9$ | $4.42428 +1.13756 \,i$ | $4.43056 +1.13432 \,i$ |
| $10$ | $4.49595 +1.12348 \,i$ | $4.50212 +1.12041 \,i$ |
| $20$ | $4.95332 +1.04533 \,i$ | $4.95880 +1.04314 \,i$ |
| $30$ | $5.21027 +1.00868 \,i$ | $5.21539 +1.00685 \,i$ |
| $40$ | $5.38829 +0.98573 \,i$ | $5.39318 +0.98411 \,i$ |
| $50$ | $5.52410 +0.96940 \,i$ | $5.52881 +0.96791 \,i$ |
| $60$ | $5.63366 +0.95690 \,i$ | $5.63824 +0.95551 \,i$ |
| $70$ | $5.72535 +0.94687 \,i$ | $5.72982 +0.94556 \,i$ |
| $80$ | $5.80411 +0.93855 \,i$ | $5.80849 +0.93730 \,i$ |
| $90$ | $5.87307 +0.93147 \,i$ | $5.87738 +0.93027 \,i$ |
| $100$ | $5.93438 +0.92534 \,i$ | $5.93862 +0.92418 \,i$ |
| $200$ | $6.32917 +0.88910 \,i$ | $6.33302 +0.88818 \,i$ |
| $300$ | $6.55383 +0.87067 \,i$ | $6.55748 +0.86985 \,i$ |
| $400$ | $6.71067 +0.85862 \,i$ | $6.71420 +0.85787 \,i$ |
| $500$ | $6.83097 +0.84980 \,i$ | $6.83439 +0.84908 \,i$ |
| $600$ | $6.92840 +0.84289 \,i$ | $6.93176 +0.84222 \,i$ |
| $700$ | $7.01021 +0.83726 \,i$ | $7.01351 +0.83661 \,i$ |
| $800$ | $7.08066 +0.83252 \,i$ | $7.08391 +0.83189 \,i$ |
| $900$ | $7.14249 +0.82845 \,i$ | $7.14570 +0.827838 \,i$ |
| $1000$ | $7.19756 +0.82489 \,i$ | $7.20073 +0.824289 \,i$ |
