Do you have a official serial number for Lightroom? Where do you have buyed your license? LR6 doesn't been saled for several years.
Normally it's no problem to activate a LR6 installation online.
Which operating system do you use?
Adobe no longer supports TLS 1.0 and 1.1. Please check your system that you're enabled the Transport Layer Security (TLS) 1.2 protocol. For more infos take a look here:
If you're using Win 7 please upgrade your OS and install the MS Easy Fix.
Update to enable TLS 1.1 and TLS 1.2 as default secure protocols in WinHTTP in Windows (microsoft.co...
Also upgrade your browsers to enable TLS 1.2 connections.
See also here: How to resolve connection errors with Creative Suite applications (adobe.com)
Especially have a look to your HOSTS file.
My System: Intel i7-8700K - 64GB RAM - NVidia Geforce GTX 1050 Ti - Windows 10 Pro 22H2 -- LR-Classic 12.3 - Photoshop 24.5 - Nik Collection 6 - Topaz Sharpener AI