| L(s) = 1 | + 3·3-s − 20.7·5-s + 9·9-s − 6.11·11-s + 22.8·13-s − 62.1·15-s − 8.70·17-s + 75.0·19-s + 178.·23-s + 303.·25-s + 27·27-s − 122.·29-s − 102.·31-s − 18.3·33-s + 182.·37-s + 68.4·39-s − 488.·41-s − 388·43-s − 186.·45-s − 254.·47-s − 26.1·51-s + 610.·53-s + 126.·55-s + 225.·57-s + 31.2·59-s + 587.·61-s − 472.·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.85·5-s + 0.333·9-s − 0.167·11-s + 0.486·13-s − 1.06·15-s − 0.124·17-s + 0.906·19-s + 1.61·23-s + 2.42·25-s + 0.192·27-s − 0.782·29-s − 0.594·31-s − 0.0967·33-s + 0.809·37-s + 0.281·39-s − 1.85·41-s − 1.37·43-s − 0.617·45-s − 0.789·47-s − 0.0717·51-s + 1.58·53-s + 0.310·55-s + 0.523·57-s + 0.0690·59-s + 1.23·61-s − 0.901·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 20.7T + 125T^{2} \) |
| 11 | \( 1 + 6.11T + 1.33e3T^{2} \) |
| 13 | \( 1 - 22.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 8.70T + 4.91e3T^{2} \) |
| 19 | \( 1 - 75.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 178.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 122.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 102.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 182.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 488.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 388T + 7.95e4T^{2} \) |
| 47 | \( 1 + 254.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 610.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 31.2T + 2.05e5T^{2} \) |
| 61 | \( 1 - 587.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 689.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 563.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 275.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 219.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.44e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 792.T + 9.12e5T^{2} \) |
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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
−8.653402580958436185428810938005, −8.319677096156143232334935306478, −7.27236025502573271668623725658, −6.98821820855166421423174792817, −5.38645324381715263160596995438, −4.45774588233557287147427917182, −3.55859730702536447742575808505, −2.98589912574423831161014216723, −1.27564001857084658784970974901, 0,
1.27564001857084658784970974901, 2.98589912574423831161014216723, 3.55859730702536447742575808505, 4.45774588233557287147427917182, 5.38645324381715263160596995438, 6.98821820855166421423174792817, 7.27236025502573271668623725658, 8.319677096156143232334935306478, 8.653402580958436185428810938005
