My desktop computer won’t turn on. I don’t know the serial number.

We don't want your serial number, that is personal information and should NOT be shared on a public forum. You can find the SN on the bottom of the computer's stand.

Please follow the instructions in If your Mac won't turn on

The reset SMC procedure for your iMac is once it is shutdown, unplug it from power, wait 10 seconds, and then reconnect to power. Power it on. Since you are not using High Sierra with an iMac with a T2 chip, you do not need to press hold any power button for this process.

The serial number is on the bottom of the iMac foot. You may need a bright light source to see it clearly.

It kind of depends on the actual computer you have, but try this.

unplug from power and hold the power button for 30 seconds

plug in power and hold for 30 seconds or until you see 