| L(s) = 1 | − 2-s + 4-s − 5-s − 1.19·7-s − 8-s + 10-s + 3.64·11-s + 4.86·13-s + 1.19·14-s + 16-s − 3.19·17-s + 2.27·19-s − 20-s − 3.64·22-s + 2.86·23-s + 25-s − 4.86·26-s − 1.19·28-s − 6.06·29-s + 5.36·31-s − 32-s + 3.19·34-s + 1.19·35-s + 37-s − 2.27·38-s + 40-s + 8.75·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.450·7-s − 0.353·8-s + 0.316·10-s + 1.09·11-s + 1.35·13-s + 0.318·14-s + 0.250·16-s − 0.774·17-s + 0.521·19-s − 0.223·20-s − 0.776·22-s + 0.598·23-s + 0.200·25-s − 0.954·26-s − 0.225·28-s − 1.12·29-s + 0.963·31-s − 0.176·32-s + 0.547·34-s + 0.201·35-s + 0.164·37-s − 0.369·38-s + 0.158·40-s + 1.36·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.252678769\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.252678769\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + 1.19T + 7T^{2} \) |
| 11 | \( 1 - 3.64T + 11T^{2} \) |
| 13 | \( 1 - 4.86T + 13T^{2} \) |
| 17 | \( 1 + 3.19T + 17T^{2} \) |
| 19 | \( 1 - 2.27T + 19T^{2} \) |
| 23 | \( 1 - 2.86T + 23T^{2} \) |
| 29 | \( 1 + 6.06T + 29T^{2} \) |
| 31 | \( 1 - 5.36T + 31T^{2} \) |
| 41 | \( 1 - 8.75T + 41T^{2} \) |
| 43 | \( 1 + 6.75T + 43T^{2} \) |
| 47 | \( 1 + 4.13T + 47T^{2} \) |
| 53 | \( 1 + 5.88T + 53T^{2} \) |
| 59 | \( 1 + 7.28T + 59T^{2} \) |
| 61 | \( 1 - 14.5T + 61T^{2} \) |
| 67 | \( 1 + 4.90T + 67T^{2} \) |
| 71 | \( 1 + 5.38T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 - 7.38T + 79T^{2} \) |
| 83 | \( 1 + 4.41T + 83T^{2} \) |
| 89 | \( 1 + 9.63T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−8.678477192735905801799787117619, −8.032654015623805802776434422897, −7.13114392831936220113820293080, −6.49015156191513113367381977847, −5.93673125883930980215055182745, −4.69358193313141894089764923850, −3.75697525328792306958560876014, −3.11970214206761240646152488649, −1.77668608694996462372195969161, −0.76905522154442237512605290149,
0.76905522154442237512605290149, 1.77668608694996462372195969161, 3.11970214206761240646152488649, 3.75697525328792306958560876014, 4.69358193313141894089764923850, 5.93673125883930980215055182745, 6.49015156191513113367381977847, 7.13114392831936220113820293080, 8.032654015623805802776434422897, 8.678477192735905801799787117619
