whether for infinitely many . As evidence, many provided Ramanujan's (case of weight 12). The only solutions up to 1010 to the equation are 2, 3, 5, 7, 2411, and . conjectured that for all , an assertion sometimes known as Lehmer's conjecture. Lehmer verified the conjecture for up to (Apostol 1997, p. 22). The following table summarizes progress on finding successively larger values of for which this condition holds for all . Ramanujan's L-function Ramanujan's L-function is defined by if and by analytic continuation otherwise. It satisfies the functional equation and has the Euler product Ramanujan conjectured that all nontrivial zeros of have real part equal to . Notes References Modular forms Multiplicative functions Srinivasa Ramanujan Zeta and L-functions