| L(s) = 1 | − 0.725·2-s + 3-s − 1.47·4-s − 3.68·5-s − 0.725·6-s − 0.281·7-s + 2.52·8-s + 9-s + 2.67·10-s − 1.72·11-s − 1.47·12-s + 13-s + 0.204·14-s − 3.68·15-s + 1.11·16-s + 1.29·17-s − 0.725·18-s − 6.38·19-s + 5.43·20-s − 0.281·21-s + 1.24·22-s + 2.49·23-s + 2.52·24-s + 8.58·25-s − 0.725·26-s + 27-s + 0.414·28-s + ⋯ |
| L(s) = 1 | − 0.513·2-s + 0.577·3-s − 0.736·4-s − 1.64·5-s − 0.296·6-s − 0.106·7-s + 0.891·8-s + 0.333·9-s + 0.845·10-s − 0.519·11-s − 0.425·12-s + 0.277·13-s + 0.0545·14-s − 0.951·15-s + 0.279·16-s + 0.314·17-s − 0.171·18-s − 1.46·19-s + 1.21·20-s − 0.0613·21-s + 0.266·22-s + 0.520·23-s + 0.514·24-s + 1.71·25-s − 0.142·26-s + 0.192·27-s + 0.0783·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 0.725T + 2T^{2} \) |
| 5 | \( 1 + 3.68T + 5T^{2} \) |
| 7 | \( 1 + 0.281T + 7T^{2} \) |
| 11 | \( 1 + 1.72T + 11T^{2} \) |
| 17 | \( 1 - 1.29T + 17T^{2} \) |
| 19 | \( 1 + 6.38T + 19T^{2} \) |
| 23 | \( 1 - 2.49T + 23T^{2} \) |
| 29 | \( 1 - 6.21T + 29T^{2} \) |
| 31 | \( 1 + 0.971T + 31T^{2} \) |
| 37 | \( 1 - 3.09T + 37T^{2} \) |
| 41 | \( 1 - 3.77T + 41T^{2} \) |
| 43 | \( 1 - 9.32T + 43T^{2} \) |
| 47 | \( 1 - 0.568T + 47T^{2} \) |
| 53 | \( 1 - 7.23T + 53T^{2} \) |
| 59 | \( 1 - 3.57T + 59T^{2} \) |
| 61 | \( 1 - 3.08T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 + 3.79T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 + 0.650T + 83T^{2} \) |
| 89 | \( 1 - 0.273T + 89T^{2} \) |
| 97 | \( 1 - 5.21T + 97T^{2} \) |
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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
−8.216300625891554525738445407860, −7.59761308080160105350199666261, −7.02622234702763353494123905446, −5.87626476784919529519972537784, −4.63882979571003214090318494154, −4.30582078242103141420033267045, −3.51458386015517987898483400727, −2.60405042300446308881283296896, −1.07612840099976588617230992299, 0,
1.07612840099976588617230992299, 2.60405042300446308881283296896, 3.51458386015517987898483400727, 4.30582078242103141420033267045, 4.63882979571003214090318494154, 5.87626476784919529519972537784, 7.02622234702763353494123905446, 7.59761308080160105350199666261, 8.216300625891554525738445407860
