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"This program doesn't seem to slow down neither my computer nor my downloads."

Is this correct, or should neither/nor be used instead?
And how about this one:

"This program doesn't seem to either slow down my computer or make my downloads go any slower."
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"This program doesn't seem to slow down either my computer or my downloads."

This one is too awkward for words.
"This program doesn't seem to either slow down my computer or make my downloads go any slower."

Try this for an alternate using neither/nor:
This program slows down neither my computer nor my downloads.
Comments  
"Neither does this program seem to slow down my computer, nor my downloads."
Exple: Neither does this memorandum place additional legal obligations on either organisation, nor imply any transfer of responsibility from one to the other, nor sharing and/or delegation of functions.

Source: http://www.healthcarecommission.org.uk/_db/_documents/BMA_MoU.pdf
Also, I think it should be "does not" + "either/or" or "does" + "neither/nor" but not "does not" + "neither/nor".
 RayH's reply was promoted to an answer.