| L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s + 7-s + 8-s + 9-s − 2·10-s − 2·11-s − 12-s + 4·13-s + 14-s + 2·15-s + 16-s + 17-s + 18-s − 19-s − 2·20-s − 21-s − 2·22-s + 5·23-s − 24-s − 25-s + 4·26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.603·11-s − 0.288·12-s + 1.10·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.229·19-s − 0.447·20-s − 0.218·21-s − 0.426·22-s + 1.04·23-s − 0.204·24-s − 1/5·25-s + 0.784·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13566 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13566 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.467631321\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.467631321\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−16.17838305352161, −15.55073049530798, −15.13846549784458, −14.71360853081742, −13.74662696188916, −13.38746326047745, −12.86940670901140, −12.11770511489904, −11.65598903706582, −11.29297637250337, −10.63940510003501, −10.20249300510165, −9.285894791728911, −8.288822610884475, −8.129477438851033, −7.285856328752133, −6.666161738705257, −6.045204639279258, −5.338536444356501, −4.721009378305771, −4.146993465544695, −3.426227578165297, −2.712872538315780, −1.576850277648754, −0.6723204736812491,
0.6723204736812491, 1.576850277648754, 2.712872538315780, 3.426227578165297, 4.146993465544695, 4.721009378305771, 5.338536444356501, 6.045204639279258, 6.666161738705257, 7.285856328752133, 8.129477438851033, 8.288822610884475, 9.285894791728911, 10.20249300510165, 10.63940510003501, 11.29297637250337, 11.65598903706582, 12.11770511489904, 12.86940670901140, 13.38746326047745, 13.74662696188916, 14.71360853081742, 15.13846549784458, 15.55073049530798, 16.17838305352161
