| L(s) = 1 | + 2-s + 3·3-s + 4-s − 5-s + 3·6-s + 7-s + 8-s + 6·9-s − 10-s + 3·12-s + 5·13-s + 14-s − 3·15-s + 16-s + 7·17-s + 6·18-s − 7·19-s − 20-s + 3·21-s + 8·23-s + 3·24-s + 25-s + 5·26-s + 9·27-s + 28-s − 29-s − 3·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 1/2·4-s − 0.447·5-s + 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s − 0.316·10-s + 0.866·12-s + 1.38·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s + 1.69·17-s + 1.41·18-s − 1.60·19-s − 0.223·20-s + 0.654·21-s + 1.66·23-s + 0.612·24-s + 1/5·25-s + 0.980·26-s + 1.73·27-s + 0.188·28-s − 0.185·29-s − 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.784689456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.784689456\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 139 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−14.76061005216174, −14.39850128752700, −13.82295986042779, −13.19649174964842, −12.84765534809062, −12.59060380765443, −11.57290076479898, −11.25213642127423, −10.42378015918388, −10.25028783641153, −9.108929099677116, −8.943844819073736, −8.350186154371873, −7.888773863326684, −7.231791285138916, −6.957827233572607, −5.934239294791990, −5.474669519138382, −4.559210949429107, −3.985632488444186, −3.590265128338403, −3.066381139497002, −2.380130202993107, −1.622333302909807, −0.9866353615282121,
0.9866353615282121, 1.622333302909807, 2.380130202993107, 3.066381139497002, 3.590265128338403, 3.985632488444186, 4.559210949429107, 5.474669519138382, 5.934239294791990, 6.957827233572607, 7.231791285138916, 7.888773863326684, 8.350186154371873, 8.943844819073736, 9.108929099677116, 10.25028783641153, 10.42378015918388, 11.25213642127423, 11.57290076479898, 12.59060380765443, 12.84765534809062, 13.19649174964842, 13.82295986042779, 14.39850128752700, 14.76061005216174
