| L(s) = 1 | − 1.14·2-s − 1.67·3-s − 0.678·4-s + 3.60·5-s + 1.92·6-s + 0.890·7-s + 3.07·8-s − 0.194·9-s − 4.14·10-s + 3.69·11-s + 1.13·12-s + 4.19·13-s − 1.02·14-s − 6.04·15-s − 2.18·16-s + 6.13·17-s + 0.223·18-s + 1.87·19-s − 2.44·20-s − 1.49·21-s − 4.24·22-s + 2.33·23-s − 5.15·24-s + 8.02·25-s − 4.82·26-s + 5.35·27-s − 0.603·28-s + ⋯ |
| L(s) = 1 | − 0.812·2-s − 0.967·3-s − 0.339·4-s + 1.61·5-s + 0.786·6-s + 0.336·7-s + 1.08·8-s − 0.0647·9-s − 1.31·10-s + 1.11·11-s + 0.328·12-s + 1.16·13-s − 0.273·14-s − 1.56·15-s − 0.545·16-s + 1.48·17-s + 0.0526·18-s + 0.431·19-s − 0.547·20-s − 0.325·21-s − 0.905·22-s + 0.486·23-s − 1.05·24-s + 1.60·25-s − 0.945·26-s + 1.02·27-s − 0.114·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.223446801\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.223446801\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 1.14T + 2T^{2} \) |
| 3 | \( 1 + 1.67T + 3T^{2} \) |
| 5 | \( 1 - 3.60T + 5T^{2} \) |
| 7 | \( 1 - 0.890T + 7T^{2} \) |
| 11 | \( 1 - 3.69T + 11T^{2} \) |
| 13 | \( 1 - 4.19T + 13T^{2} \) |
| 17 | \( 1 - 6.13T + 17T^{2} \) |
| 19 | \( 1 - 1.87T + 19T^{2} \) |
| 23 | \( 1 - 2.33T + 23T^{2} \) |
| 29 | \( 1 + 5.17T + 29T^{2} \) |
| 31 | \( 1 - 6.24T + 31T^{2} \) |
| 37 | \( 1 + 1.04T + 37T^{2} \) |
| 41 | \( 1 + 1.49T + 41T^{2} \) |
| 47 | \( 1 - 7.55T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 0.0871T + 59T^{2} \) |
| 61 | \( 1 + 8.99T + 61T^{2} \) |
| 67 | \( 1 - 2.36T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 + 4.41T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 + 8.14T + 83T^{2} \) |
| 89 | \( 1 + 1.48T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−9.306725547580755619675308016389, −8.731770723021957120493833213638, −7.81366733326062419149055942822, −6.69817175067965942388172011459, −5.96798807216144736141896805265, −5.45884014836992166516546265715, −4.57580979465627768581832763182, −3.26405802609705852639057145090, −1.57326109122745137607234605100, −1.04947310905042239760553806900,
1.04947310905042239760553806900, 1.57326109122745137607234605100, 3.26405802609705852639057145090, 4.57580979465627768581832763182, 5.45884014836992166516546265715, 5.96798807216144736141896805265, 6.69817175067965942388172011459, 7.81366733326062419149055942822, 8.731770723021957120493833213638, 9.306725547580755619675308016389
