| L(s) = 1 | − 1.83·2-s + 1.36·4-s + 3.15·5-s + 2.30·7-s + 1.17·8-s − 5.79·10-s − 4.75·11-s − 6.46·13-s − 4.22·14-s − 4.86·16-s − 6.20·17-s − 1.72·19-s + 4.29·20-s + 8.71·22-s − 7.50·23-s + 4.97·25-s + 11.8·26-s + 3.13·28-s − 0.513·29-s − 1.19·31-s + 6.58·32-s + 11.3·34-s + 7.27·35-s − 1.39·37-s + 3.16·38-s + 3.70·40-s − 6.48·41-s + ⋯ |
| L(s) = 1 | − 1.29·2-s + 0.680·4-s + 1.41·5-s + 0.871·7-s + 0.414·8-s − 1.83·10-s − 1.43·11-s − 1.79·13-s − 1.12·14-s − 1.21·16-s − 1.50·17-s − 0.396·19-s + 0.961·20-s + 1.85·22-s − 1.56·23-s + 0.995·25-s + 2.32·26-s + 0.592·28-s − 0.0954·29-s − 0.214·31-s + 1.16·32-s + 1.95·34-s + 1.23·35-s − 0.228·37-s + 0.513·38-s + 0.585·40-s − 1.01·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 1.83T + 2T^{2} \) |
| 5 | \( 1 - 3.15T + 5T^{2} \) |
| 7 | \( 1 - 2.30T + 7T^{2} \) |
| 11 | \( 1 + 4.75T + 11T^{2} \) |
| 13 | \( 1 + 6.46T + 13T^{2} \) |
| 17 | \( 1 + 6.20T + 17T^{2} \) |
| 19 | \( 1 + 1.72T + 19T^{2} \) |
| 23 | \( 1 + 7.50T + 23T^{2} \) |
| 29 | \( 1 + 0.513T + 29T^{2} \) |
| 31 | \( 1 + 1.19T + 31T^{2} \) |
| 37 | \( 1 + 1.39T + 37T^{2} \) |
| 41 | \( 1 + 6.48T + 41T^{2} \) |
| 43 | \( 1 - 1.09T + 43T^{2} \) |
| 47 | \( 1 - 6.86T + 47T^{2} \) |
| 53 | \( 1 - 1.24T + 53T^{2} \) |
| 59 | \( 1 + 9.79T + 59T^{2} \) |
| 61 | \( 1 + 1.04T + 61T^{2} \) |
| 67 | \( 1 - 4.24T + 67T^{2} \) |
| 71 | \( 1 - 5.53T + 71T^{2} \) |
| 73 | \( 1 + 5.11T + 73T^{2} \) |
| 79 | \( 1 - 9.28T + 79T^{2} \) |
| 83 | \( 1 - 8.19T + 83T^{2} \) |
| 89 | \( 1 - 18.0T + 89T^{2} \) |
| 97 | \( 1 - 4.11T + 97T^{2} \) |
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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
−9.658760732390348031819821434699, −8.957334345756207979946971222242, −8.047705964752946413434286685032, −7.44930221333324497563448562679, −6.39059886422810444654934384953, −5.21111432800931205596634514133, −4.62494586478415278746032954491, −2.23078831272491942332105727808, −2.07742047510289357217721050579, 0,
2.07742047510289357217721050579, 2.23078831272491942332105727808, 4.62494586478415278746032954491, 5.21111432800931205596634514133, 6.39059886422810444654934384953, 7.44930221333324497563448562679, 8.047705964752946413434286685032, 8.957334345756207979946971222242, 9.658760732390348031819821434699
