| L(s) = 1 | + 2.02·2-s − 0.589·3-s + 2.10·4-s − 5-s − 1.19·6-s − 0.911·7-s + 0.215·8-s − 2.65·9-s − 2.02·10-s + 11-s − 1.24·12-s − 6.64·13-s − 1.84·14-s + 0.589·15-s − 3.77·16-s − 5.09·17-s − 5.37·18-s + 19-s − 2.10·20-s + 0.537·21-s + 2.02·22-s + 7.09·23-s − 0.127·24-s + 25-s − 13.4·26-s + 3.33·27-s − 1.92·28-s + ⋯ |
| L(s) = 1 | + 1.43·2-s − 0.340·3-s + 1.05·4-s − 0.447·5-s − 0.487·6-s − 0.344·7-s + 0.0762·8-s − 0.884·9-s − 0.640·10-s + 0.301·11-s − 0.358·12-s − 1.84·13-s − 0.493·14-s + 0.152·15-s − 0.943·16-s − 1.23·17-s − 1.26·18-s + 0.229·19-s − 0.470·20-s + 0.117·21-s + 0.432·22-s + 1.47·23-s − 0.0259·24-s + 0.200·25-s − 2.64·26-s + 0.641·27-s − 0.362·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.02T + 2T^{2} \) |
| 3 | \( 1 + 0.589T + 3T^{2} \) |
| 7 | \( 1 + 0.911T + 7T^{2} \) |
| 13 | \( 1 + 6.64T + 13T^{2} \) |
| 17 | \( 1 + 5.09T + 17T^{2} \) |
| 23 | \( 1 - 7.09T + 23T^{2} \) |
| 29 | \( 1 + 0.220T + 29T^{2} \) |
| 31 | \( 1 + 5.38T + 31T^{2} \) |
| 37 | \( 1 - 9.78T + 37T^{2} \) |
| 41 | \( 1 + 6.38T + 41T^{2} \) |
| 43 | \( 1 - 0.507T + 43T^{2} \) |
| 47 | \( 1 - 7.55T + 47T^{2} \) |
| 53 | \( 1 + 1.04T + 53T^{2} \) |
| 59 | \( 1 - 6.15T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 - 5.35T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 + 7.42T + 73T^{2} \) |
| 79 | \( 1 + 3.78T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 - 7.72T + 97T^{2} \) |
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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
−9.422511049999361926944644145955, −8.807439687281848247144769402184, −7.43428094330836272248453240431, −6.77504998140759298150240098480, −5.87393645899999298471467169262, −4.98498719936313158923783310824, −4.43136759321867475075959173546, −3.20462174674664952749673212950, −2.47952896234910799644114122845, 0,
2.47952896234910799644114122845, 3.20462174674664952749673212950, 4.43136759321867475075959173546, 4.98498719936313158923783310824, 5.87393645899999298471467169262, 6.77504998140759298150240098480, 7.43428094330836272248453240431, 8.807439687281848247144769402184, 9.422511049999361926944644145955
