| L(s) = 1 | + 2.14·2-s + 2.60·4-s − 2.30·5-s + 1.71·7-s + 1.30·8-s − 4.94·10-s − 11-s − 5.20·13-s + 3.67·14-s − 2.41·16-s − 2.40·17-s − 5.97·19-s − 6.00·20-s − 2.14·22-s + 23-s + 0.307·25-s − 11.1·26-s + 4.46·28-s − 5.57·29-s + 6.55·31-s − 7.79·32-s − 5.15·34-s − 3.94·35-s − 0.451·37-s − 12.8·38-s − 3.00·40-s − 0.0529·41-s + ⋯ |
| L(s) = 1 | + 1.51·2-s + 1.30·4-s − 1.03·5-s + 0.647·7-s + 0.460·8-s − 1.56·10-s − 0.301·11-s − 1.44·13-s + 0.983·14-s − 0.604·16-s − 0.582·17-s − 1.36·19-s − 1.34·20-s − 0.457·22-s + 0.208·23-s + 0.0615·25-s − 2.18·26-s + 0.844·28-s − 1.03·29-s + 1.17·31-s − 1.37·32-s − 0.884·34-s − 0.667·35-s − 0.0742·37-s − 2.07·38-s − 0.474·40-s − 0.00826·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2277 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2277 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 2.14T + 2T^{2} \) |
| 5 | \( 1 + 2.30T + 5T^{2} \) |
| 7 | \( 1 - 1.71T + 7T^{2} \) |
| 13 | \( 1 + 5.20T + 13T^{2} \) |
| 17 | \( 1 + 2.40T + 17T^{2} \) |
| 19 | \( 1 + 5.97T + 19T^{2} \) |
| 29 | \( 1 + 5.57T + 29T^{2} \) |
| 31 | \( 1 - 6.55T + 31T^{2} \) |
| 37 | \( 1 + 0.451T + 37T^{2} \) |
| 41 | \( 1 + 0.0529T + 41T^{2} \) |
| 43 | \( 1 - 6.61T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 1.67T + 53T^{2} \) |
| 59 | \( 1 + 0.751T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 - 4.92T + 67T^{2} \) |
| 71 | \( 1 - 4.97T + 71T^{2} \) |
| 73 | \( 1 - 4.33T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 + 7.20T + 89T^{2} \) |
| 97 | \( 1 + 0.228T + 97T^{2} \) |
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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
−8.437821098099281105574416360300, −7.62415643461190048265110461333, −7.03361331714382684887953074910, −6.08631760969086317116138151911, −5.19809700911460628503687560683, −4.41415079810112398528826875884, −4.12739405520044860020741282036, −2.89389397621718255751563942071, −2.12912243375545003101016481759, 0,
2.12912243375545003101016481759, 2.89389397621718255751563942071, 4.12739405520044860020741282036, 4.41415079810112398528826875884, 5.19809700911460628503687560683, 6.08631760969086317116138151911, 7.03361331714382684887953074910, 7.62415643461190048265110461333, 8.437821098099281105574416360300
