| L(s) = 1 | − 2.63·2-s − 3-s + 4.95·4-s − 2.22·5-s + 2.63·6-s − 2.31·7-s − 7.77·8-s + 9-s + 5.85·10-s + 1.95·11-s − 4.95·12-s + 1.32·13-s + 6.10·14-s + 2.22·15-s + 10.6·16-s − 2.63·18-s + 5.95·19-s − 11.0·20-s + 2.31·21-s − 5.14·22-s − 4.77·23-s + 7.77·24-s − 0.0618·25-s − 3.48·26-s − 27-s − 11.4·28-s + 0.172·29-s + ⋯ |
| L(s) = 1 | − 1.86·2-s − 0.577·3-s + 2.47·4-s − 0.993·5-s + 1.07·6-s − 0.874·7-s − 2.75·8-s + 0.333·9-s + 1.85·10-s + 0.588·11-s − 1.42·12-s + 0.366·13-s + 1.63·14-s + 0.573·15-s + 2.65·16-s − 0.621·18-s + 1.36·19-s − 2.45·20-s + 0.505·21-s − 1.09·22-s − 0.996·23-s + 1.58·24-s − 0.0123·25-s − 0.683·26-s − 0.192·27-s − 2.16·28-s + 0.0320·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 2.63T + 2T^{2} \) |
| 5 | \( 1 + 2.22T + 5T^{2} \) |
| 7 | \( 1 + 2.31T + 7T^{2} \) |
| 11 | \( 1 - 1.95T + 11T^{2} \) |
| 13 | \( 1 - 1.32T + 13T^{2} \) |
| 19 | \( 1 - 5.95T + 19T^{2} \) |
| 23 | \( 1 + 4.77T + 23T^{2} \) |
| 29 | \( 1 - 0.172T + 29T^{2} \) |
| 31 | \( 1 - 0.444T + 31T^{2} \) |
| 37 | \( 1 - 5.31T + 37T^{2} \) |
| 41 | \( 1 - 4.02T + 41T^{2} \) |
| 43 | \( 1 - 8.33T + 43T^{2} \) |
| 47 | \( 1 - 2.38T + 47T^{2} \) |
| 53 | \( 1 + 0.727T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 8.68T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 - 6.10T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 0.698T + 79T^{2} \) |
| 83 | \( 1 - 8.52T + 83T^{2} \) |
| 89 | \( 1 - 5.40T + 89T^{2} \) |
| 97 | \( 1 + 4.71T + 97T^{2} \) |
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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
−9.534639637450347741994477058492, −9.162920760571702702964465027452, −7.906511030281942882583206895888, −7.53914063026429437279342239185, −6.54034687535028564658088524423, −5.87713530491430579319659946947, −4.07035346802347767371359104758, −2.92761943191645360938725656432, −1.24432532923392769496882383606, 0,
1.24432532923392769496882383606, 2.92761943191645360938725656432, 4.07035346802347767371359104758, 5.87713530491430579319659946947, 6.54034687535028564658088524423, 7.53914063026429437279342239185, 7.906511030281942882583206895888, 9.162920760571702702964465027452, 9.534639637450347741994477058492
