| L(s) = 1 | − 2.21·2-s + 2.92·4-s + 0.739·5-s + 7-s − 2.05·8-s − 1.64·10-s − 1.02·11-s + 1.47·13-s − 2.21·14-s − 1.28·16-s + 3.12·17-s − 5.88·19-s + 2.16·20-s + 2.26·22-s + 1.53·23-s − 4.45·25-s − 3.27·26-s + 2.92·28-s + 1.52·29-s + 1.14·31-s + 6.97·32-s − 6.93·34-s + 0.739·35-s − 11.1·37-s + 13.0·38-s − 1.52·40-s + 11.0·41-s + ⋯ |
| L(s) = 1 | − 1.56·2-s + 1.46·4-s + 0.330·5-s + 0.377·7-s − 0.727·8-s − 0.518·10-s − 0.308·11-s + 0.409·13-s − 0.593·14-s − 0.321·16-s + 0.758·17-s − 1.34·19-s + 0.483·20-s + 0.483·22-s + 0.320·23-s − 0.890·25-s − 0.642·26-s + 0.553·28-s + 0.283·29-s + 0.205·31-s + 1.23·32-s − 1.19·34-s + 0.124·35-s − 1.82·37-s + 2.11·38-s − 0.240·40-s + 1.72·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 2.21T + 2T^{2} \) |
| 5 | \( 1 - 0.739T + 5T^{2} \) |
| 11 | \( 1 + 1.02T + 11T^{2} \) |
| 13 | \( 1 - 1.47T + 13T^{2} \) |
| 17 | \( 1 - 3.12T + 17T^{2} \) |
| 19 | \( 1 + 5.88T + 19T^{2} \) |
| 23 | \( 1 - 1.53T + 23T^{2} \) |
| 29 | \( 1 - 1.52T + 29T^{2} \) |
| 31 | \( 1 - 1.14T + 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 1.60T + 43T^{2} \) |
| 47 | \( 1 + 8.29T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 + 3.45T + 59T^{2} \) |
| 61 | \( 1 - 5.01T + 61T^{2} \) |
| 67 | \( 1 + 8.41T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 + 1.36T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 + 7.52T + 83T^{2} \) |
| 89 | \( 1 - 1.31T + 89T^{2} \) |
| 97 | \( 1 + 8.68T + 97T^{2} \) |
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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
−7.66421846994623311128061860707, −7.07901913268441876967132812703, −6.28656976746025128306567106328, −5.64119304904860698505004579625, −4.69811122042059223325230090424, −3.83234304506555712855872281464, −2.69178046499684449809200014512, −1.92572808639951301453056912690, −1.16154690007489232643764181575, 0,
1.16154690007489232643764181575, 1.92572808639951301453056912690, 2.69178046499684449809200014512, 3.83234304506555712855872281464, 4.69811122042059223325230090424, 5.64119304904860698505004579625, 6.28656976746025128306567106328, 7.07901913268441876967132812703, 7.66421846994623311128061860707
