For the syrup: Combine 1 1/4 cups water and the sugar in a small pot. Bring to a boil. Boil for 5 minutes, and then reduce the heat and let simmer for 15 minutes. Remove from the heat, stir in the lemon juice and pour into another bowl so that it cools down quickly. For the baklava: Preheat the oven to 360 degrees F. Brush a baking sheet with butter. Unroll one package of phyllo. Lay 2 sheets of phyllo down in the center of the baking sheet, and then drizzle with some butter. Repeat the layering 4 more times, each time allowing 2 to 3 inches of the phyllo to drape over one rim of the pan, until all 4 rims of the pan have been covered. Sprinkle with a thin layer of pistachios. Fold the draped phyllo over the pistachio and drizzle with more butter. Repeat the layering with the phyllo, butter and remaining pistachios until the first package of phyllo (24 sheets total) is used up.
Unroll the second package of phyllo. Lay 2 sheets of phyllo dough onto the previous layers of baklava. Drizzle lightly with butter. Repeat layering phyllo and drizzling with butter until the second package of phyllo is finished. Dip a sharp knife into hot water and cut the baklava into rectangles. Brush the top liberally with butter.
Bake the baklava for 15 minutes on a low rack in the oven, and then transfer to the top rack and bake for 15 minutes longer. For best taste, cool and let rest for 24 hours so that the syrup fully permeates the baklava.