Thonky.com's Pokémon Legends: Z-A Walkthrough

L's memory has partly returned, and he asks you to find two Pokémon, including "the one that takes".

Talk to Grisham

Go to Nouveau Café and approach the truck. Afterward, go into the Quasartico Inc. building. Then when you are ready, talk to Vinnie, and defeat each of the simulated rogue Mega Pokémon (Victreebel, Hawlucha, and Tyranitar).

Afterward, Yveltal will be on the roof of the Galerie de la Lune. To reach Yveltal, go around the northeast side of the buildings and go up the ramps, then fall down below the narrow plank between the scaffolding. Go northwest from there to find a ladder. Climb up it, then use Roto-Glide to reach the ladder nearby. Use Roto-Glide to get onto the dome, and check on the mission destination. Then use Roto-Glide to get onto the dome again, and check on Yveltal to start the battle. You can throw Poké Balls at it throughout the battle, so keep trying. Timer Balls will become more effective the longer the battle goes on.

After you complete this mission and Main Mission 40: The One That Gives, try to go into Hotel Z, and you will start the mission Main Mission 42: To Keep the World in Balance.