Linux: zip greater than 4Gb

The basic Zip format has a limit of 4 GB per file. You need to upgrade your zip tool to one that supports Zip64:

$ zip -v
Copyright (c) 1990-2008 Info-ZIP - Type 'zip "-L"' for software license.
This is Zip 3.0 (July 5th 2008), by Info-ZIP.
...

Zip special compilation options:
        ...
        ZIP64_SUPPORT        (use Zip64 to store large files in archives)

Alternatively, use a different archive format such as 7z or tar.

use the linux split command to chop your log file into smaller files.

and consider setting up log rotate so this doesn't happen again.