| L(s) = 1 | − 2-s + 2.23·3-s + 4-s + 0.346·5-s − 2.23·6-s + 1.27·7-s − 8-s + 2.00·9-s − 0.346·10-s − 4.01·11-s + 2.23·12-s − 1.58·13-s − 1.27·14-s + 0.774·15-s + 16-s + 4.76·17-s − 2.00·18-s − 19-s + 0.346·20-s + 2.84·21-s + 4.01·22-s − 3.55·23-s − 2.23·24-s − 4.88·25-s + 1.58·26-s − 2.22·27-s + 1.27·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.29·3-s + 0.5·4-s + 0.154·5-s − 0.913·6-s + 0.480·7-s − 0.353·8-s + 0.667·9-s − 0.109·10-s − 1.20·11-s + 0.645·12-s − 0.439·13-s − 0.339·14-s + 0.200·15-s + 0.250·16-s + 1.15·17-s − 0.472·18-s − 0.229·19-s + 0.0774·20-s + 0.620·21-s + 0.855·22-s − 0.741·23-s − 0.456·24-s − 0.976·25-s + 0.310·26-s − 0.428·27-s + 0.240·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 2.23T + 3T^{2} \) |
| 5 | \( 1 - 0.346T + 5T^{2} \) |
| 7 | \( 1 - 1.27T + 7T^{2} \) |
| 11 | \( 1 + 4.01T + 11T^{2} \) |
| 13 | \( 1 + 1.58T + 13T^{2} \) |
| 17 | \( 1 - 4.76T + 17T^{2} \) |
| 23 | \( 1 + 3.55T + 23T^{2} \) |
| 29 | \( 1 + 5.97T + 29T^{2} \) |
| 31 | \( 1 - 5.18T + 31T^{2} \) |
| 37 | \( 1 - 7.85T + 37T^{2} \) |
| 41 | \( 1 - 3.23T + 41T^{2} \) |
| 43 | \( 1 + 12.4T + 43T^{2} \) |
| 47 | \( 1 + 2.70T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 - 1.46T + 59T^{2} \) |
| 61 | \( 1 - 0.631T + 61T^{2} \) |
| 67 | \( 1 + 4.91T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 + 9.52T + 73T^{2} \) |
| 79 | \( 1 + 2.20T + 79T^{2} \) |
| 83 | \( 1 + 2.94T + 83T^{2} \) |
| 89 | \( 1 + 18.2T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−7.60841779370843533591913164045, −7.36857353986171329177668163983, −6.08146771200636474159757320685, −5.53276090040751747760928647963, −4.57953702762925238957377390535, −3.64213293334153813081837855177, −2.85511450774194715035402351435, −2.26156685914355343397513763065, −1.48964382956637716524617843279, 0,
1.48964382956637716524617843279, 2.26156685914355343397513763065, 2.85511450774194715035402351435, 3.64213293334153813081837855177, 4.57953702762925238957377390535, 5.53276090040751747760928647963, 6.08146771200636474159757320685, 7.36857353986171329177668163983, 7.60841779370843533591913164045
