Can I read the e-book that I found via Finna.fi search service?

Sure, you can ask.

You can read the e-book that you found via the Finna.fi search service with your JAMK network account if the e-book has been purchased to JAMK Library’s collection. Meaning, we have not purchased all the e-books that can be found on Finna and e-books purchased by other libraries cannot unfortunately be read using a JAMK network account. Note, however, that there are also open access e-books on Finna.

You can see from the drop-down menu if JAMK Library has purchased the e-book: