The Lizard-Spock Expansion

The first mention occurs in Season 2, Episode 8, "The Lizard-Spock Expansion," at the very beginning of the episode:

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SHELDON: Oh look, Saturn 3 is on.
RAJ: I don't want to watch Saturn 3. Deep Space Nine is better.

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SHELDON: How is Deep Space Nine better than Saturn 3?

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RAJ: Simple subtraction will tell you it's six better.

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LEONARD: Compromise. Watch Babylon 5.

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SHELDON: In what sense is that a compromise?

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LEONARD: Well, five is partway between three – never mind.


The Vegas Renormalization

The second mention occurs later in Season 2, Episode 21, "The Vegas Renormalization," at 1:47. The gang is eating supper and going around the circle taking turns thinking of a character, and the others have to guess. It's Howard's turn, and he's just returned after receiving a phone call from Leslie Winkle:

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LEONARD: Okay, let's see. Are you from a TV series?

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HOWARD: She dumped me!

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SHELDON: I bet he's someone from Babylon 5, we're never going to get it.


The Large Hadron Collision

The third mention occurs in Season 3, Episode 15, "The Large Hadron Collision," at 11:50. Sheldon has made breakfast in bed for Leonard as an apology:

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SHELDON: After you've finished breakfast, I thought we could spend the day watching the final season of Babylon 5 with director commentary.

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LEONARD: You hate Babylon 5.

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SHELDON: I do. It fails as drama, science fiction, and it's hopelessly derivative. But you like it, and you're my friend.


The Staircase Implementation

The fourth mention occurs later in Season 3, Episode 22, "The Staircase Implementation," in a flashback scene at 13:27. Leonard has just moved into the apartment with Sheldon:

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LEONARD: Ooh, it's time for Babylon 5!

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SHELDON: We don't watch Babylon 5 in this apartment.
LEONARD: Why not?

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SHELDON: Because no one likes Babylon 5.

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LEONARD: I like it.

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RAJ: Me too.
HOWARD: So do I.

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LEONARD: There you go, three against one.

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SHELDON: They don't get a vote. It's one against one. And according to the roommate agreement, all ties will be settled by me.

LEONARD: But I said no to that.

SHELDON: And I said yes. And I settle all ties.

The fifth mention also occurs in Season 3, Episode 22, "The Staircase Implementation," at 19:22 in the final scene of the episode:

INT. SHELDON AND LEONARD'S APARTMENT

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LEONARD: Never mind. Okay if I watch some TV?
SHELDON: Go ahead.

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TV VOICE: Up next: Babylon 5.

INT. PENNY'S APARTMENT

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LEONARD (O.S.): You're not even watching!
SHELDON (O.S.): I can hear it!
LEONARD (O.S.): Oh, so the dialogue offends you?
SHELDON (O.S.): I would hardly call that dialogue!

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LEONARD (O.S.): You are insane, you know that?!
SHELDON (O.S.): Don't make me turn that flag upside down, 'cause you know I'll do it!


The Hawking Excitation

The sixth mention occurs in Season 5, Episode 21, "The Hawking Excitation" at 4:08. After Howard tells the group how his day with Stephen Hawking went, Sheldon begs him for the opportunity to meet Hawking:

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SHELDON: Howard, please, I'm begging you.

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LEONARD: Raj, you're our group historian. Has Sheldon ever begged before?

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RAJ: Three times. He begged the Fox network not to cancel Firefly. He begged the TNT network to cancel Babylon 5. And when he got food poisoning at the Rose Bowl Parade, he begged a deity he doesn't believe in to end his life quickly.