First, reserve about 1/4 cup of the sugar and set aside. Pour the rest of the sugar into a medium saucepan. Add one cup of water. Stir and bring to a boil to dissolve sugar (this is called simple syrup). Turn off heat and let cool a bit.
Meanwhile, pour the reserved 1/4 cup of sugar over the mashed raspberries and let them sit for at least ten minutes to macerate.
While waiting for the berries, zest 6 of the 7 lemons. Place the zest into a zipper sandwich bag, squeeze out all the air, and put it in the freezer. Never cut up a lemon and waste the zest! :) Cut the six lemons in half and extract the juice (this should yield a little under a cup of juice). Cut the 7th lemon into slices. Set aside.
Pour the lemon juice through a strainer to eliminate the pulp and any seeds. Pour the juice into a 2 quart pitcher.
Push the macerated berries through a strainer using the back of a spoon. Pour the sweetened berry juice into the pitcher with the lemon juice.
Pour in the simple syrup (sugar water) into the pitcher and top off with water. Stir, add sliced lemons, and refrigerate.