That's because they were removed from power supplies with atx12v spec version 1.3 and above. Seasonic 620 is based on v 2.3. Pretty sure that 17th pin was also made for 20 pin power supplies- the psu for very VERY VERY old systems. That is not the reason why the system is not starting.
To check if the power supply is able to power up on its own, follow these instructions exactly:
How To: Jump Start A Power Supply (PSU) / Test A Power Supply And Components
If the fan on the psu or if you've connected any on the the psu spins, that means that the power supply is able to supply power. If it doesn't, rma the power supply. If it does, we can rule out the power supply for the time being.