| L(s) = 1 | − 4.75·2-s + 14.6·4-s + 7.65·5-s − 31.5·7-s − 31.6·8-s − 36.4·10-s + 7.18·11-s + 84.3·13-s + 150.·14-s + 33.5·16-s − 17·17-s − 37.0·19-s + 112.·20-s − 34.2·22-s − 150.·23-s − 66.3·25-s − 401.·26-s − 462.·28-s + 11.5·29-s − 53.2·31-s + 93.7·32-s + 80.9·34-s − 241.·35-s − 99.2·37-s + 176.·38-s − 242.·40-s − 118.·41-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 1.83·4-s + 0.684·5-s − 1.70·7-s − 1.40·8-s − 1.15·10-s + 0.197·11-s + 1.79·13-s + 2.87·14-s + 0.524·16-s − 0.242·17-s − 0.447·19-s + 1.25·20-s − 0.331·22-s − 1.36·23-s − 0.531·25-s − 3.02·26-s − 3.12·28-s + 0.0741·29-s − 0.308·31-s + 0.517·32-s + 0.408·34-s − 1.16·35-s − 0.440·37-s + 0.753·38-s − 0.958·40-s − 0.450·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 17 | \( 1 + 17T \) |
| good | 2 | \( 1 + 4.75T + 8T^{2} \) |
| 5 | \( 1 - 7.65T + 125T^{2} \) |
| 7 | \( 1 + 31.5T + 343T^{2} \) |
| 11 | \( 1 - 7.18T + 1.33e3T^{2} \) |
| 13 | \( 1 - 84.3T + 2.19e3T^{2} \) |
| 19 | \( 1 + 37.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 150.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 11.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 53.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 99.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 118.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 456.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 571.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 462.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 48.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 59.5T + 2.26e5T^{2} \) |
| 67 | \( 1 + 740.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 930.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 697.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 22.2T + 5.71e5T^{2} \) |
| 89 | \( 1 - 369.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.13e3T + 9.12e5T^{2} \) |
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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
−11.65644661908976076971321866031, −10.50400401685843674619413223268, −9.838401726963965250614617301965, −9.068366168463588903802152142660, −8.142182592671096268013463040669, −6.56917714299354251397802095120, −6.17627826640342648198481797484, −3.47203228415881083087051326614, −1.75646561566549658193007977758, 0,
1.75646561566549658193007977758, 3.47203228415881083087051326614, 6.17627826640342648198481797484, 6.56917714299354251397802095120, 8.142182592671096268013463040669, 9.068366168463588903802152142660, 9.838401726963965250614617301965, 10.50400401685843674619413223268, 11.65644661908976076971321866031
