| L(s) = 1 | + 2.24·2-s + 3.03·4-s − 3.18·5-s − 3.85·7-s + 2.33·8-s − 7.15·10-s + 2.63·11-s + 0.540·13-s − 8.64·14-s − 0.838·16-s − 7.97·17-s − 5.14·19-s − 9.68·20-s + 5.90·22-s + 2.63·23-s + 5.15·25-s + 1.21·26-s − 11.7·28-s + 0.903·29-s − 1.00·31-s − 6.55·32-s − 17.9·34-s + 12.2·35-s − 4.62·37-s − 11.5·38-s − 7.43·40-s + 6.90·41-s + ⋯ |
| L(s) = 1 | + 1.58·2-s + 1.51·4-s − 1.42·5-s − 1.45·7-s + 0.825·8-s − 2.26·10-s + 0.793·11-s + 0.149·13-s − 2.31·14-s − 0.209·16-s − 1.93·17-s − 1.18·19-s − 2.16·20-s + 1.25·22-s + 0.548·23-s + 1.03·25-s + 0.238·26-s − 2.21·28-s + 0.167·29-s − 0.179·31-s − 1.15·32-s − 3.07·34-s + 2.07·35-s − 0.760·37-s − 1.87·38-s − 1.17·40-s + 1.07·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 - 2.24T + 2T^{2} \) |
| 5 | \( 1 + 3.18T + 5T^{2} \) |
| 7 | \( 1 + 3.85T + 7T^{2} \) |
| 11 | \( 1 - 2.63T + 11T^{2} \) |
| 13 | \( 1 - 0.540T + 13T^{2} \) |
| 17 | \( 1 + 7.97T + 17T^{2} \) |
| 19 | \( 1 + 5.14T + 19T^{2} \) |
| 23 | \( 1 - 2.63T + 23T^{2} \) |
| 29 | \( 1 - 0.903T + 29T^{2} \) |
| 31 | \( 1 + 1.00T + 31T^{2} \) |
| 37 | \( 1 + 4.62T + 37T^{2} \) |
| 41 | \( 1 - 6.90T + 41T^{2} \) |
| 43 | \( 1 - 3.17T + 43T^{2} \) |
| 47 | \( 1 + 5.57T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 9.44T + 59T^{2} \) |
| 61 | \( 1 + 4.85T + 61T^{2} \) |
| 67 | \( 1 - 0.118T + 67T^{2} \) |
| 71 | \( 1 - 9.55T + 71T^{2} \) |
| 73 | \( 1 - 0.637T + 73T^{2} \) |
| 79 | \( 1 + 3.93T + 79T^{2} \) |
| 83 | \( 1 + 4.23T + 83T^{2} \) |
| 89 | \( 1 - 2.93T + 89T^{2} \) |
| 97 | \( 1 - 2.65T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.222208605016291835663179353881, −8.656116143651013390902176284146, −7.28240281299690271331759264107, −6.61764850720684333805396792474, −6.14961670092252570245998917744, −4.71622107037810075981176061252, −4.05610452163208676686040957274, −3.50359222501151620365752927498, −2.46812594998313440111618342912, 0,
2.46812594998313440111618342912, 3.50359222501151620365752927498, 4.05610452163208676686040957274, 4.71622107037810075981176061252, 6.14961670092252570245998917744, 6.61764850720684333805396792474, 7.28240281299690271331759264107, 8.656116143651013390902176284146, 9.222208605016291835663179353881
