| L(s) = 1 | + 1.62·2-s − 5.34·4-s − 14.4·5-s − 35.3·7-s − 21.7·8-s − 23.5·10-s + 7.37·11-s + 34.6·13-s − 57.6·14-s + 7.32·16-s + 34.7·17-s + 43.9·19-s + 77.3·20-s + 12.0·22-s + 65.6·23-s + 84.3·25-s + 56.5·26-s + 189.·28-s − 82.7·29-s + 267.·31-s + 185.·32-s + 56.6·34-s + 511.·35-s + 81.1·37-s + 71.6·38-s + 314.·40-s − 489.·41-s + ⋯ |
| L(s) = 1 | + 0.576·2-s − 0.668·4-s − 1.29·5-s − 1.91·7-s − 0.960·8-s − 0.745·10-s + 0.202·11-s + 0.739·13-s − 1.10·14-s + 0.114·16-s + 0.495·17-s + 0.530·19-s + 0.864·20-s + 0.116·22-s + 0.594·23-s + 0.674·25-s + 0.426·26-s + 1.27·28-s − 0.530·29-s + 1.54·31-s + 1.02·32-s + 0.285·34-s + 2.47·35-s + 0.360·37-s + 0.305·38-s + 1.24·40-s − 1.86·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 - 1.62T + 8T^{2} \) |
| 5 | \( 1 + 14.4T + 125T^{2} \) |
| 7 | \( 1 + 35.3T + 343T^{2} \) |
| 11 | \( 1 - 7.37T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 34.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 43.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 65.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 82.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 267.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 81.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 489.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 385.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 398.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 41.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 127.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 380.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 473.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 763.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 693.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 399.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 73.3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.11e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 811.T + 9.12e5T^{2} \) |
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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
−8.439808588547143053574290012977, −7.50984355147040951049076037605, −6.62663481211911996067883921967, −6.01461309303150916343823014551, −5.01262936656143243836973118935, −4.03481819288381258869942342838, −3.44092220654966566267604010444, −3.00992713159256065874062979576, −0.853468566508426915397320774919, 0,
0.853468566508426915397320774919, 3.00992713159256065874062979576, 3.44092220654966566267604010444, 4.03481819288381258869942342838, 5.01262936656143243836973118935, 6.01461309303150916343823014551, 6.62663481211911996067883921967, 7.50984355147040951049076037605, 8.439808588547143053574290012977
