…So i bought a 5000k lamp and try to manage a direct comparison between prints and displayed image…
The difference in colors is less than it is perceived in the photos.
The light of lamp is not illuminating directly the photos.
Up to now, I haven’t printed any of my work, but I am looking forward to doing so. If you review a print under a lamp in your work area, but the area where the print will eventually hang has different light characteristics, is that a problem?
In general, no.
Professional print shops use light booths to check their stuff in light conditions that are super rare in the real world. But it is a stable reference point to work with, just like the light on your desk or viewing area.
You face a much worse problem with every picture that you send out digitally. Screens and viewing conditions change so much and you have no control over that.
Also humans are very good at adapting to different light and ignoring color casts. So without the reference most are pretty much lost and everything looks great for them. Which is a good thing.
If the light characteristics of the viewing area are really extreme and you know that beforehand, than you should of course take that into consideration or let it fall under “not my problem”. What do I mean by extreme? A red lamp. Black light. Stuff like that. A few 100 Kelvin either direction don’t really justify the extra work.