| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 2·7-s + 8-s + 9-s − 10-s − 12-s + 13-s − 2·14-s + 15-s + 16-s − 6·17-s + 18-s − 4·19-s − 20-s + 2·21-s − 2·23-s − 24-s + 25-s + 26-s − 27-s − 2·28-s + 30-s − 8·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.277·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.436·21-s − 0.417·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.377·28-s + 0.182·30-s − 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.98272218073684, −12.70191707772891, −12.15198825998944, −11.81453803193303, −11.15605331712164, −10.97870517636627, −10.51499294600196, −10.03566362858316, −9.400344780541114, −8.909583912741083, −8.572930254794526, −7.821817260867848, −7.370799091718465, −6.907463658881356, −6.428004102172755, −6.063801515282187, −5.694472335242342, −4.878271917950337, −4.564384773093103, −4.067973145618690, −3.567669043338597, −3.090022331802599, −2.304473698473995, −1.915322509208955, −1.113536007169391, 0, 0,
1.113536007169391, 1.915322509208955, 2.304473698473995, 3.090022331802599, 3.567669043338597, 4.067973145618690, 4.564384773093103, 4.878271917950337, 5.694472335242342, 6.063801515282187, 6.428004102172755, 6.907463658881356, 7.370799091718465, 7.821817260867848, 8.572930254794526, 8.909583912741083, 9.400344780541114, 10.03566362858316, 10.51499294600196, 10.97870517636627, 11.15605331712164, 11.81453803193303, 12.15198825998944, 12.70191707772891, 12.98272218073684
