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Can the death ray fire in a 360 degree arc around the Doomsycthe since the codex says you can place the first point anywhere within 12" of the gun mount? Are there any restrictions due to line of sight or can this just be ignored due to the codex entry rule?

You can ignore LOS for the purposes of placing the deathray initially. That said this is rendered rather pointless because you cannot actually 'wound' a model you don't have LOS on, so putting the line so it hits anything that is out of LOS will have no practical effect.

RAW you can draw the line, determine hits and create a wound pool regardless of LOS from the weapon.

However, since the weapon is a fixed weapon, any unit that has a wound pool generated must be in LOS along the weapon barrel to actually allocate the wounds.

I would not be surprised if this gets FAQ'd within the next decade or so.