To be pretty blunt... be glad you didn't get a permaban. Riot has a zero-tolerance policy with regards to hate speech. And whether you were mad at someone or not, dropping a homophobic slur three times in a row definitely qualifies as hate speech. While you might know that you weren't mad, the rest of your team, and the enemy team has no way of knowing that. To anyone else who reads your chat, it simply looks like you are slurring the enemy team for not following through on an open-mid.
If you haven't received your email by tomorrow (sometimes it can take up to 24 hours and be sure to check your spam/junk folders just in case it was sent there by mistake), you can always submit a ticket to Player Support and ask them to send you the logs via ticket response or resend the email. Support will be more than happy to assist you with that matter, and they can usually get back to you within 24-48 hours, though I've always found their response time to be closer to 3-4 hours.
One last thing you may want to keep in mind, the punishment system works off of tiers. If a player earns additionally punishments within a short period of time after the previous punishment, they are bumped up to the next tier. In the case of severe toxicity, or certain behaviors which Riot considers to be extremely inappropriate (typically hate speech, threats, things of that nature) punishment can be escalated to skip tiers. There are four tiers of punishment.
Tier 1: 10 game chat restriction
Tier 2: 25 game chat restriction
Tier 3: 14 day temporary suspension
Tier 4: permanent suspension
You are at Tier 3. At this stage, your next punishment is a permanent suspension. And it doesn't necessarily take much for a player to be bumped that last tier. I've seen it said a few times that it's actually easier for a player to go from a 14 day suspension to a permaban than it is for a player to go from nothing to their first chat restriction. And just from the logs that I've seen of various players here, it seems to be true. Try to be careful. It would be a shame to lose your account over something minor, when it can so easily be avoided. Just keep your head down, try to refrain from using any slurs, and as long as you show positive behavior, your punishment tier will drop down to where you won't be instantly permabanned on your next punishment. Which hopefully, since you've come here before it's too late, will not happen. Best of luck in the future.