| L(s) = 1 | − 0.621·2-s − 1.61·4-s − 0.389·5-s + 0.149·7-s + 2.24·8-s + 0.241·10-s − 5.16·11-s + 4.31·13-s − 0.0931·14-s + 1.83·16-s − 1.54·17-s + 5.99·19-s + 0.628·20-s + 3.20·22-s + 2.02·23-s − 4.84·25-s − 2.68·26-s − 0.242·28-s + 3.55·29-s − 8.03·31-s − 5.62·32-s + 0.962·34-s − 0.0583·35-s − 5.37·37-s − 3.72·38-s − 0.873·40-s − 2.95·41-s + ⋯ |
| L(s) = 1 | − 0.439·2-s − 0.807·4-s − 0.173·5-s + 0.0566·7-s + 0.793·8-s + 0.0764·10-s − 1.55·11-s + 1.19·13-s − 0.0248·14-s + 0.458·16-s − 0.375·17-s + 1.37·19-s + 0.140·20-s + 0.684·22-s + 0.421·23-s − 0.969·25-s − 0.525·26-s − 0.0457·28-s + 0.659·29-s − 1.44·31-s − 0.995·32-s + 0.164·34-s − 0.00986·35-s − 0.884·37-s − 0.604·38-s − 0.138·40-s − 0.461·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 2 | \( 1 + 0.621T + 2T^{2} \) |
| 5 | \( 1 + 0.389T + 5T^{2} \) |
| 7 | \( 1 - 0.149T + 7T^{2} \) |
| 11 | \( 1 + 5.16T + 11T^{2} \) |
| 13 | \( 1 - 4.31T + 13T^{2} \) |
| 17 | \( 1 + 1.54T + 17T^{2} \) |
| 19 | \( 1 - 5.99T + 19T^{2} \) |
| 23 | \( 1 - 2.02T + 23T^{2} \) |
| 29 | \( 1 - 3.55T + 29T^{2} \) |
| 31 | \( 1 + 8.03T + 31T^{2} \) |
| 37 | \( 1 + 5.37T + 37T^{2} \) |
| 41 | \( 1 + 2.95T + 41T^{2} \) |
| 43 | \( 1 + 1.22T + 43T^{2} \) |
| 47 | \( 1 - 1.80T + 47T^{2} \) |
| 53 | \( 1 + 5.44T + 53T^{2} \) |
| 59 | \( 1 - 1.46T + 59T^{2} \) |
| 61 | \( 1 - 3.92T + 61T^{2} \) |
| 67 | \( 1 + 5.35T + 67T^{2} \) |
| 71 | \( 1 + 0.634T + 71T^{2} \) |
| 73 | \( 1 - 8.66T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 9.05T + 97T^{2} \) |
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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
−9.196465851384847880123705651719, −8.341764145109545549115823265567, −7.86334041330612542450201063565, −6.97725906225722884464977369201, −5.61641876458864172691658523982, −5.11839578473478127917261945004, −3.99101521701413287763962148712, −3.06278151396145064108757725270, −1.49284707899769701526054384380, 0,
1.49284707899769701526054384380, 3.06278151396145064108757725270, 3.99101521701413287763962148712, 5.11839578473478127917261945004, 5.61641876458864172691658523982, 6.97725906225722884464977369201, 7.86334041330612542450201063565, 8.341764145109545549115823265567, 9.196465851384847880123705651719
