| L(s) = 1 | − 1.53·2-s + 0.370·4-s + 1.28·5-s + 3.45·7-s + 2.50·8-s − 1.97·10-s − 11-s + 1.86·13-s − 5.32·14-s − 4.60·16-s − 6.66·17-s + 5.03·19-s + 0.476·20-s + 1.53·22-s − 0.946·23-s − 3.34·25-s − 2.86·26-s + 1.28·28-s + 1.98·29-s − 4.90·31-s + 2.07·32-s + 10.2·34-s + 4.44·35-s + 3.26·37-s − 7.75·38-s + 3.22·40-s + 0.871·41-s + ⋯ |
| L(s) = 1 | − 1.08·2-s + 0.185·4-s + 0.575·5-s + 1.30·7-s + 0.886·8-s − 0.626·10-s − 0.301·11-s + 0.516·13-s − 1.42·14-s − 1.15·16-s − 1.61·17-s + 1.15·19-s + 0.106·20-s + 0.328·22-s − 0.197·23-s − 0.669·25-s − 0.562·26-s + 0.242·28-s + 0.367·29-s − 0.881·31-s + 0.366·32-s + 1.76·34-s + 0.751·35-s + 0.536·37-s − 1.25·38-s + 0.509·40-s + 0.136·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 1.53T + 2T^{2} \) |
| 5 | \( 1 - 1.28T + 5T^{2} \) |
| 7 | \( 1 - 3.45T + 7T^{2} \) |
| 13 | \( 1 - 1.86T + 13T^{2} \) |
| 17 | \( 1 + 6.66T + 17T^{2} \) |
| 19 | \( 1 - 5.03T + 19T^{2} \) |
| 23 | \( 1 + 0.946T + 23T^{2} \) |
| 29 | \( 1 - 1.98T + 29T^{2} \) |
| 31 | \( 1 + 4.90T + 31T^{2} \) |
| 37 | \( 1 - 3.26T + 37T^{2} \) |
| 41 | \( 1 - 0.871T + 41T^{2} \) |
| 43 | \( 1 - 0.489T + 43T^{2} \) |
| 47 | \( 1 + 0.630T + 47T^{2} \) |
| 53 | \( 1 + 2.31T + 53T^{2} \) |
| 59 | \( 1 + 9.46T + 59T^{2} \) |
| 61 | \( 1 + 13.2T + 61T^{2} \) |
| 67 | \( 1 + 4.39T + 67T^{2} \) |
| 71 | \( 1 + 7.10T + 71T^{2} \) |
| 73 | \( 1 - 5.03T + 73T^{2} \) |
| 79 | \( 1 + 2.48T + 79T^{2} \) |
| 83 | \( 1 + 8.79T + 83T^{2} \) |
| 89 | \( 1 + 17.5T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−7.68404710022508679699824620144, −7.10125114318129194359683971479, −6.15153628992796197810356604554, −5.40398849735459501467837688299, −4.65537839730190116026618334362, −4.11828768748044208657365970446, −2.78327336090028396664360343181, −1.80077752271656319791182811942, −1.35686808498374276663015708009, 0,
1.35686808498374276663015708009, 1.80077752271656319791182811942, 2.78327336090028396664360343181, 4.11828768748044208657365970446, 4.65537839730190116026618334362, 5.40398849735459501467837688299, 6.15153628992796197810356604554, 7.10125114318129194359683971479, 7.68404710022508679699824620144
