| L(s) = 1 | + 2-s − 2.96·3-s + 4-s − 0.775·5-s − 2.96·6-s + 0.413·7-s + 8-s + 5.80·9-s − 0.775·10-s − 1.47·11-s − 2.96·12-s − 4.24·13-s + 0.413·14-s + 2.29·15-s + 16-s + 3.63·17-s + 5.80·18-s + 4.33·19-s − 0.775·20-s − 1.22·21-s − 1.47·22-s − 23-s − 2.96·24-s − 4.39·25-s − 4.24·26-s − 8.32·27-s + 0.413·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.71·3-s + 0.5·4-s − 0.346·5-s − 1.21·6-s + 0.156·7-s + 0.353·8-s + 1.93·9-s − 0.245·10-s − 0.444·11-s − 0.856·12-s − 1.17·13-s + 0.110·14-s + 0.593·15-s + 0.250·16-s + 0.881·17-s + 1.36·18-s + 0.993·19-s − 0.173·20-s − 0.267·21-s − 0.314·22-s − 0.208·23-s − 0.605·24-s − 0.879·25-s − 0.833·26-s − 1.60·27-s + 0.0782·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 + 2.96T + 3T^{2} \) |
| 5 | \( 1 + 0.775T + 5T^{2} \) |
| 7 | \( 1 - 0.413T + 7T^{2} \) |
| 11 | \( 1 + 1.47T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 - 3.63T + 17T^{2} \) |
| 19 | \( 1 - 4.33T + 19T^{2} \) |
| 29 | \( 1 - 0.921T + 29T^{2} \) |
| 31 | \( 1 + 5.94T + 31T^{2} \) |
| 37 | \( 1 - 4.03T + 37T^{2} \) |
| 41 | \( 1 - 4.31T + 41T^{2} \) |
| 43 | \( 1 + 1.20T + 43T^{2} \) |
| 47 | \( 1 - 4.26T + 47T^{2} \) |
| 53 | \( 1 + 4.51T + 53T^{2} \) |
| 59 | \( 1 - 0.667T + 59T^{2} \) |
| 61 | \( 1 - 8.50T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 + 4.55T + 71T^{2} \) |
| 73 | \( 1 + 9.18T + 73T^{2} \) |
| 79 | \( 1 - 7.14T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−7.54130084946024866102084787341, −6.90135191803128047646442847201, −6.06570873623600014761880107533, −5.42502490305325546980299394512, −5.07737135690876007411681055199, −4.30410828847511676966631317352, −3.46882582985798444148957364861, −2.33008411126637184760098530526, −1.14699789212870105172057796757, 0,
1.14699789212870105172057796757, 2.33008411126637184760098530526, 3.46882582985798444148957364861, 4.30410828847511676966631317352, 5.07737135690876007411681055199, 5.42502490305325546980299394512, 6.06570873623600014761880107533, 6.90135191803128047646442847201, 7.54130084946024866102084787341
