I have a pretty big problem with my 3 TB hard disk.

Before I used the hard disk for the first time I've overwritten the whole hard disk with shred -n 1 -v.

The hard drive has/had 5 partitions:
/dev/sda1 1 MB BIOS boot partition (type EF02) (currently not used)
/dev/sda2 150 MB /boot (ext4)
/dev/sda3 40 GB LUKS/LVM (with LVs swap, /, /home, each formatted with ext4)
/dev/sda4 150 MB /boot (ext4)
/dev/sda5 approx. 2.7 TB LUKS/LVM (with LVs swap, /, /home, wach formatted with ext4)

Salix' lilo was installed to the protective MBR, which necessary with my main board.

/dev/sda4 and /dev/sda5 were used as my main system with Salix OS. On /dev/sda2 and /dev/sda3 I wanted to install Sabayon.

So I prepared /dev/sda2 and /dev/sda3, created a new LUKS container on /dev/sda3, created a new LVM layout on this LUKS container and formatted all of these partitions with ext4.

I gave the Sabayon installer the password for this LUKS partition, but not for my main LUKS partition. The Sabayon installer could open it and showed every partition and every LVM volume correctly. So I set the mount points and told the installer not to format any of these partitions.

The bootloader (grub2) should be installed to /dev/sda2 instead of the MBR.

Then the system got installed, and after about 20 or 30 minutes I was told to reboot the system. Then the problem appeared.

The BIOS POST was finished and then it did nothing, not even an error message that no bootable media was found. I couldn't even boot from a LiveCD anymore.

After removing the hard disk I could boot again from a LiveCD.

Then I connected the hard disk with a docking station and ran gdisk -l. I currently can't give the exact output, but it showed me, that there's totally no partition on the hard disk, and not even a partition table. It looked like a totally new hard disk. It could also not find a backup of the (protective) MBR or partition table.

Gparted showed me the hard disk as unallocated.

The analysis of TestDisk showed me a lot of garbage, but one line which could remind me of /dev/sda5, which is the biggest and most important partition. At least the size in this output looked like the size of this partition. The partition type was mentioned as unknown. But TestDisk didn't show this entry later for recovery.

I don't know what crap the Sabayon installer has done, but I have the hope that the last partition is still physically on the hard disk.

My question is now: Is it possible to recover this last LUKS partition (/dev/sda5) and probably the whole GPT, hopefully without a professional laboratory which is way too expensive for me? Or if it's not possible to fix this by myself, would it make sense to give it to a professional laboratory?

And, please, no comments about backups. There is (temporarily) no backup, and this has a reason.

What would be helpful is not an account of things but cold, factual data: tool output. For example running test disk with the /debug /log switches and posting or attaching its log file contents.