The answer is "Superman's Day of Doom!", from Superman 157, cover dated November 1962 (as reprinted in Limited Collectors' Edition C-38, October-November 1975). It doesn't answer why all these characters have the same initials, but it gets a whole lot of mileage out of the fact that they do.
Yes, it's Superman Day in Metropolis! When millions - millions! - of people gather in the streets of Metropolis to pay homage to the omnipotent alien to whom they owe their lives. Don't be late for the Superman Day parade! If you are the millionth person there you will not see a thing.
Hey aliens! If you've been considering launching a sneak rocket attack on Earth but have been worried about being detected, Superman Day is the day for you. Just slap "Hooray for our best buddy Superman!" on the side of your ultimo-destructo-mega-missile and you're good to go!
Superman - whose private thoughts often include mocking laughter - receives a present from grateful space aliens, courtesy of an extremely trusting Naval officer.
"But Admiral, I TOLD him it was for Superman! If you can't trust a room full of journalists with a mysterious package, who can you trust?"
Lois and Lana tear into that wrapping paper like it's a stripper's Superman-branded thong. (Sorry.) Lois reads the enclosed note - it's a present from the Cybern Galaxy, because Superman once helped them repel an invasion. Yes, it's a present from an entire galaxy. It better be a pretty darn good present. (Spoiler - it's not.)
Uh oh, the Cybernians - everyone in the entire Cybern Galaxy - are mathematical geniuses! They've given Superman an advanced kind of computer which can predict the future, and which, like all advanced kinds of computer, only works three times. Clark, the Charlie Brown of the Daily Planet, tries to stop everybody's fun by threatening them with Superman's wrath, but Lois is determined. Lois Lane, world-famous journalist, has been presented with the chance to learn the answer to any question. Whatever will she do with this once-in-a-lifetime opportunity?
Oh, right, Silver Age Lois Lane. Pretty much a one-track mind, and that track runs one-way to Superman's super-regions. What will the predictor machine say?
L. L.! Here are those famous initials. But L.L. could mean anyone - as Lois points out, it could mean Lana Lang, or Lois Lane. Or as Superman is thinking, it could mean Lori Lemaris, a beautiful mermaid he once dated, or Linda Lee, who is the alter-ego of Supergirl, his TEENAGE COUSIN!
There was a very disturbing trend in Silver Age Superman stories of Clark being heavy into his little cousin. I know, I know, plenty of cousins get married, even first cousins, like Superman and Supergirl. But there is a serious creep factor here - she seems to be somewhere in the neighborhood of fourteen years old in the bulk of these stories, her dying parents specifically sent her to Superman so that he could raise her, and - forget what I just said - she's his first cousin! He's always going on about how beautiful she is, and if only he could find a girl like her, and what a shame it is that it's illegal by Kryptonian law for first cousins to marry (yes, he really said that).
But hey, whatever. He's just keeping his options open, and so is the predictor machine. While Supes is pondering incestuous jailbait, Jimmy jumps in to try his luck.
Good choice, Jimmy. Not a waste of a question at all - somebody give him a monkey's paw, he'll clean up. Once again, the machine answers "L L". Jimmy assumes the machine is broken, but Clark knows he's going to Las Vegas and Los Angeles - the machine must be right! Just like when John Edward asks if anyone in the audience knows someone with a J in their name - it's eerily accurate!
Meanwhile, Perry White receives a threat from "the underworld" (they've become pretty unified since Superman came to town) that they're going to bump Clark off during the Superman Day parade in retribution for a story he wrote which put some of them in jail. Perry wants to give Clark a police escort, but Clark - knowing he's going to have to duck out to attend the parade as Superman - tries to convince Perry he'll be fine on his own. He asks the predictor machine who will save him if he gets into trouble, assuming the answer will be "Superman". But of course...
L.L. again! How eerie! Confused as to why his buddies the Cybernians would send him a predictor machine that is so vague in its responses as to be functionally useless (he phrases it more politely than that), Superman heads out on one last patrol before the parade starts.
Uh-oh! Tembo's at it again! And the entire zoo is abandoned so that everyone can attend the mandatory Superman Day parade!
Given that in the last panel we saw Tembo crushing the cage that was designed to hold him with his trunk, I'm not sure a couple of iron rails in the dirt are going to keep him in. I think I could push those over, I doubt the giant elephant will have any trouble. But whatever, it killed two panels.
He mentally chuckles a lot, doesn't he? I guess things are slow with the entire city attending the parade, so Superman swoops down to put the equipment away. No problem's too small for Superman!
Why, it's Bizarro! What a pleasant surprise! And he has a present for you, Superman! I'm sure there's absolutely nothing to worry about. Why would there be? It's Bizarro! He's your pal!
Yes, on Bizarro World, pain is pleasure and torment is joy - their S&M clubs would rock your little world apart. Bizarro brought green kryptonite, the only substance that can kill Superman (except for magic, and Doomsday, and anything that would kill a normal person if he's under a red sun, and Carrot Top - Superman LOVES prop comedy). Superman's helpless, and with all of Metropolis' millions of people at the parade, there's nobody around to save him. He's doomed!
As he lies in the dust in agony, Superman remembers the predictor machine, and his hopes rise. His thoughts turn one by one to the numerous people in his life with the initials L. L., convinced that one of them will save him. He starts by trying to send a message to his telepathic mermaid ex-girlfriend (we've all got one, am I right, guys?), Lori Lemaris.
Lori's in the parade, being carried along in a fishtank just barely bigger than her body, being gawped at by Joe and Jane Average and their kids. Not demeaning at all. But will she hear him?
Even if it weren't for the interference, I don't recall Lori having "immunity to electricity" as a super-power. I hope the trick was worth it, it was a one-shot deal. Maybe she was counting on a bigger tank.
Next up - Superman's fellow super-hero Lighting Lad, from the Legion of Super-Heroes! The Legion has traveled back 1000 years in time to attend the Superman Day parade! (Do you think they just do all the Superman Days in a row, in one trip, to get them out of the way?) Superman sees Lighting Lad hurling lightning bolts in his direction, and his spirits soar...
That was a pretty good shot, considering he's miles away. Surely the next one will shatter the kryptonite into thousands of smaller pieces, which is apparently a good thing for some reason!
Yes, it was mere coincidence that Lightning Lad's bolts struck so close to Superman. He was just helping out a photographer by hurling violently destructive bolts of lightning randomly across the city of Metropolis. Anything for a fan!
Next, Superman sees Lois and Lana in the world's smallest helicopter flying towards him.
Those bitches! After all you've done for them! Fire up a blast of your heat vision, Superman, see who's laughing then. Oh, wait, there's probably some suitably absurd Silver Age explanation...
Lois...you're really going to dangle on a rope from a helicopter just to untangle a parade balloon? One-handed? Without a safety harness? That's...pretty bad-ass. Of course, Lois is probably assuming that if she falls, Superman will catch her, just like he's done the previous ten thousand times she's plummeted from a great height.
"Yes, maybe my hot young cousin Linda Lee is the L.L. who will save me! I can feel her vibrating the earth below me. Let me just fine-tune my X-ray vision a little..."
What a bitter and ironic twist of fate! Supergirl was just burrowing below the earth's crust to gather enough diamonds to fuse into a statue of Superman! Nope, nothing creepy about their relationship at all.
Luma Lynai is the Superwoman of another planet. Superman fell in love with her because she looks like a grown-up version of his cousin Supergirl. Yeah.
He must be really desperate if he's looking to Lex Luthor for help. I do like how casual Luthor is about his whole predicament. The writers could have said he was in prison, but no. He's stuck on a robot world for some reason. There was no story explaining how he got there, and when we next see him there'll be no explanation for how he escaped, but he's cool, he's chilling. It's the Silver Age, sometimes you're gonna be a prisoner on a robot world, what are you gonna do?
Just as things look hopeless for the Man of Steel, an errant fly ball comes bouncing his way, followed by some random kid. He's saved!
Superman assumes that his boy savior must have the initials L.L. like the predictor machine predicted, but no...it's Steven Snapinn, which is a real name apparently. But the machine couldn't be wrong, could it?
And this is why Superman should never go to a psychic. If the kid wasn't a little leaguer Superman would have come up with some other justification - "He's a Lively Lad!" "He Looks Lost!" "He Loves Licorice!" "He's Like Lois, but smarter!" "He's in the Last stage of Leukemia!" This is how they get you, people! They let you fill in your own blanks!
And no, Superman imagining a "fitting reward" has nothing to do with the fact that the boy is bending over in front of him. You're thinking of Batman.
"Hurray!" say the ruffians in the bleachers. The rich folks in the front offer polite applause. Three cheers for Steven Snapinn, whose entire life from here on would be wasted in futile attempts to recapture the glory of this one moment! Hurray!