| L(s) = 1 | − 3.87·5-s + 2.18·7-s + 0.162·11-s + 2.41·13-s − 3·17-s − 3.59·19-s + 2.83·23-s + 10.0·25-s − 6.71·29-s + 5.18·31-s − 8.47·35-s − 6.63·37-s + 5.80·41-s + 6.22·43-s + 7.39·47-s − 2.22·49-s − 1.40·53-s − 0.630·55-s − 5.12·59-s − 3.78·61-s − 9.35·65-s + 5.86·67-s − 15.3·71-s + 8.68·73-s + 0.355·77-s + 1.26·79-s − 8.47·83-s + ⋯ |
| L(s) = 1 | − 1.73·5-s + 0.825·7-s + 0.0489·11-s + 0.668·13-s − 0.727·17-s − 0.825·19-s + 0.591·23-s + 2.00·25-s − 1.24·29-s + 0.931·31-s − 1.43·35-s − 1.09·37-s + 0.905·41-s + 0.949·43-s + 1.07·47-s − 0.318·49-s − 0.192·53-s − 0.0850·55-s − 0.666·59-s − 0.484·61-s − 1.16·65-s + 0.716·67-s − 1.81·71-s + 1.01·73-s + 0.0404·77-s + 0.142·79-s − 0.930·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 3.87T + 5T^{2} \) |
| 7 | \( 1 - 2.18T + 7T^{2} \) |
| 11 | \( 1 - 0.162T + 11T^{2} \) |
| 13 | \( 1 - 2.41T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + 3.59T + 19T^{2} \) |
| 23 | \( 1 - 2.83T + 23T^{2} \) |
| 29 | \( 1 + 6.71T + 29T^{2} \) |
| 31 | \( 1 - 5.18T + 31T^{2} \) |
| 37 | \( 1 + 6.63T + 37T^{2} \) |
| 41 | \( 1 - 5.80T + 41T^{2} \) |
| 43 | \( 1 - 6.22T + 43T^{2} \) |
| 47 | \( 1 - 7.39T + 47T^{2} \) |
| 53 | \( 1 + 1.40T + 53T^{2} \) |
| 59 | \( 1 + 5.12T + 59T^{2} \) |
| 61 | \( 1 + 3.78T + 61T^{2} \) |
| 67 | \( 1 - 5.86T + 67T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 - 8.68T + 73T^{2} \) |
| 79 | \( 1 - 1.26T + 79T^{2} \) |
| 83 | \( 1 + 8.47T + 83T^{2} \) |
| 89 | \( 1 + 7.72T + 89T^{2} \) |
| 97 | \( 1 + 3.90T + 97T^{2} \) |
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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
−8.129306675461022116651624390610, −7.47121996593214277545433940460, −6.85504095659697584647232344179, −5.89022826964615820466444961335, −4.83341215912606419559855530729, −4.23953675827492579621695492413, −3.65736091024836392054639909659, −2.57177888177779657641788038413, −1.28298563183084351914027469446, 0,
1.28298563183084351914027469446, 2.57177888177779657641788038413, 3.65736091024836392054639909659, 4.23953675827492579621695492413, 4.83341215912606419559855530729, 5.89022826964615820466444961335, 6.85504095659697584647232344179, 7.47121996593214277545433940460, 8.129306675461022116651624390610
