Labrys and Aigis were part of a band of orphans with Mana and two other boys, Keita and Musashi (who you've done art of before).
They got abducted for an experiment; Labrys and Aigis volunteered to be the first test subjects, so as to spare the other three.
Long story short, Labrys's entire being is bound within the silver gem in the axe, whilst Aigis's entire being is bound within the red gem in the shield. Through other complicated paranormal science, they themselves became bound — as weapons — to Mana Kirishima.
However, as the plot has progress, it's clear that they're still 'present', to an extent. Whether they'll ever be free of those gems? Well, who knows.
So Labrys isn't dead. Not sure if her current state of existence counts as being 'alive', though...