You might get any of a really wide range of responses.
I think the best bet is to email / send a letter (in a -handwritten- envelope) asking them to contact you about a family matter from XXX number of years ago. Keep it neutral and don't lay it all out at once, that's too much.
If they take up contact mention that you believe that you have a parent in common, that you aren't looking for any gain from them and that you just wish for some degree of contact.
Take it from there gently. It could be a big shock to them.
Be prepared for rejection though. It hurts like holy hell but there is NOTHING you can do if they don't want to know. Shield your heart if you can .. it's difficult but might help.
If they do want contact then take it slowly and don't expect too much. At best you've great new half-siblings, but it's fairly rare that it works out that well.
All the very best. It's worked out reasonably well for me on one side, and badly on the other; totally rejected. On the side that accepted me, they knew I existed first. On the other side, they didnt, and it was socially unacceptable in their circle to have an illegitimate child. It hurts. On the other hand my half sister is a glowing light in my life.