I send a email to inquire about a research internship 3 weeks ago to a professor leading a research group(they clearly state that they have on their web-page) and I like the research group's research area a lot, since my own field now is close to theirs. But I acted stupid in the email that I didn't mentioned anything about their work, just saying that I am doing research in a similar field(although I have read their papers, I didn't recognize the importance of talking about it by then due to inexperience). And the professor didn't respond me. I guess he just ignored it as a SPAM. Do you guys recommend me send a followup email to make up for it? Or it is no use reach him twice that he would merely consider it an annoyance?
It is quite possible the email just got lost in the flood. Some professors, especially the more famous ones, get up to 100 emails a day - not including the spam. It is also possible that the professor is waiting for all the applications to come in, or for some funding for the intern research costs to be decided, or just didn't get round to deciding what this summer's intern projects will be.
I would recommend you write a slightly more detailed email (don't go crazy, but a bit about why you want the internship and how it fits with your studies/projects, the kind of things you identify in your email that you perhaps could have written). You can frame it as "I thought I should provide more information" rather than "I want to bug you".
The other alternative is to look for clues about who might be the actual supervisor in reality - in a large group, interns often work for PhD students, postdocs or associate professors. If the research internships have their titles on the webpage, do some sleuthing and write to the most relevant person. Or if there are associate professors or other experienced researchers in that same group, pick one who does something interesting and write to them. Good luck!