More investigation isn’t necessary. Low T combined with low normal LH and FSH and with the symptoms you describe point to a pituitary origin for the low T. Normally if T is low, then LH and FSH will be high attempting to compensate - which would indicate a testicle problem, but this isn’t your case.
I would first try clomid to see if you can stimulate the LH production and restore normal T levels. If clomid doesn’t work, then you need to keep an eye on your pituitary for further hormone loss, which may come over the next few years (GH would normally be next after LH and FSH loss, followed by thyroid and coritsol). Meanwhile, its time to start testosterone replacement because low T has a lot of bad long term side effects, either with androgel or shots.
Personally I find androgel annoying to put on every day. The shots are not as easy but once weekly is a lot easier to live with over time.
I FOUND THIS ON SOMEONE WHO IS SUFFERING WITH PROSTATE CANCER THIS WAS ON ANOTHER FORM