| L(s) = 1 | − 2.78·3-s − 3.92·5-s + 3.56·7-s + 4.76·9-s + 5.87·11-s + 0.0875·13-s + 10.9·15-s − 6.04·17-s − 0.151·19-s − 9.93·21-s + 0.889·23-s + 10.4·25-s − 4.92·27-s + 1.61·29-s − 7.24·31-s − 16.3·33-s − 13.9·35-s − 9.00·37-s − 0.243·39-s + 2.42·41-s − 8.83·43-s − 18.7·45-s − 7.65·47-s + 5.70·49-s + 16.8·51-s + 4.15·53-s − 23.0·55-s + ⋯ |
| L(s) = 1 | − 1.60·3-s − 1.75·5-s + 1.34·7-s + 1.58·9-s + 1.77·11-s + 0.0242·13-s + 2.82·15-s − 1.46·17-s − 0.0346·19-s − 2.16·21-s + 0.185·23-s + 2.08·25-s − 0.948·27-s + 0.299·29-s − 1.30·31-s − 2.84·33-s − 2.36·35-s − 1.48·37-s − 0.0390·39-s + 0.378·41-s − 1.34·43-s − 2.79·45-s − 1.11·47-s + 0.814·49-s + 2.36·51-s + 0.571·53-s − 3.10·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 113 | \( 1 - T \) |
| good | 3 | \( 1 + 2.78T + 3T^{2} \) |
| 5 | \( 1 + 3.92T + 5T^{2} \) |
| 7 | \( 1 - 3.56T + 7T^{2} \) |
| 11 | \( 1 - 5.87T + 11T^{2} \) |
| 13 | \( 1 - 0.0875T + 13T^{2} \) |
| 17 | \( 1 + 6.04T + 17T^{2} \) |
| 19 | \( 1 + 0.151T + 19T^{2} \) |
| 23 | \( 1 - 0.889T + 23T^{2} \) |
| 29 | \( 1 - 1.61T + 29T^{2} \) |
| 31 | \( 1 + 7.24T + 31T^{2} \) |
| 37 | \( 1 + 9.00T + 37T^{2} \) |
| 41 | \( 1 - 2.42T + 41T^{2} \) |
| 43 | \( 1 + 8.83T + 43T^{2} \) |
| 47 | \( 1 + 7.65T + 47T^{2} \) |
| 53 | \( 1 - 4.15T + 53T^{2} \) |
| 59 | \( 1 - 14.0T + 59T^{2} \) |
| 61 | \( 1 + 7.06T + 61T^{2} \) |
| 67 | \( 1 - 9.06T + 67T^{2} \) |
| 71 | \( 1 - 5.56T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 + 3.75T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 1.70T + 89T^{2} \) |
| 97 | \( 1 + 4.19T + 97T^{2} \) |
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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
−7.31635607907003501969554157064, −6.84586720544207159340320430077, −6.38593596688550690444662241826, −5.12639535835094291147927203442, −4.87800470537716077871305880635, −4.03189618315812396313353025243, −3.68398460613943416285549351261, −1.88557968583821121240078861378, −1.00237625939428000212216755961, 0,
1.00237625939428000212216755961, 1.88557968583821121240078861378, 3.68398460613943416285549351261, 4.03189618315812396313353025243, 4.87800470537716077871305880635, 5.12639535835094291147927203442, 6.38593596688550690444662241826, 6.84586720544207159340320430077, 7.31635607907003501969554157064
