L(s) = 1 | + 3i·3-s − 6·9-s − 2·11-s + 6i·13-s + 2i·17-s − 9i·23-s − 9i·27-s − 3·29-s − 2·31-s − 6i·33-s − 8i·37-s − 18·39-s − 5·41-s + i·43-s + 8i·47-s + ⋯ |
L(s) = 1 | + 1.73i·3-s − 2·9-s − 0.603·11-s + 1.66i·13-s + 0.485i·17-s − 1.87i·23-s − 1.73i·27-s − 0.557·29-s − 0.359·31-s − 1.04i·33-s − 1.31i·37-s − 2.88·39-s − 0.780·41-s + 0.152i·43-s + 1.16i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 3iT - 3T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 9iT - 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 - 3iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - iT - 83T^{2} \) |
| 89 | \( 1 - 13T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.408277140958806228287753518600, −7.50823272196090852894644699960, −6.49176688220893051341645388940, −5.86334114230815019830181054007, −4.90052384919199256924302417166, −4.43501428389432261230717684267, −3.83076256145652786175862968559, −2.89881391484758588961314145583, −1.95080776127910395710579502641, 0,
1.05937551446751380999745845803, 1.95607256066154315170377205674, 2.90512876067443649451119459585, 3.50515946983528528627303331144, 5.18324546372668551716939047869, 5.47384510050324402071798148042, 6.33551367007462144395110013984, 7.08874548742258553351475943474, 7.75259099431473841333950284663