| L(s) = 1 | + 3-s − 4.38·5-s + 9-s − 11-s − 3.17·13-s − 4.38·15-s − 5.71·17-s + 6.24·19-s + 8.93·23-s + 14.2·25-s + 27-s + 8.29·29-s − 0.636·31-s − 33-s − 8.79·37-s − 3.17·39-s + 2.64·41-s − 4.11·43-s − 4.38·45-s + 5.85·47-s − 5.71·51-s − 0.386·53-s + 4.38·55-s + 6.24·57-s − 14.1·59-s + 11.7·61-s + 13.9·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.96·5-s + 0.333·9-s − 0.301·11-s − 0.881·13-s − 1.13·15-s − 1.38·17-s + 1.43·19-s + 1.86·23-s + 2.84·25-s + 0.192·27-s + 1.54·29-s − 0.114·31-s − 0.174·33-s − 1.44·37-s − 0.509·39-s + 0.413·41-s − 0.626·43-s − 0.653·45-s + 0.854·47-s − 0.800·51-s − 0.0530·53-s + 0.591·55-s + 0.826·57-s − 1.84·59-s + 1.49·61-s + 1.72·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 4.38T + 5T^{2} \) |
| 13 | \( 1 + 3.17T + 13T^{2} \) |
| 17 | \( 1 + 5.71T + 17T^{2} \) |
| 19 | \( 1 - 6.24T + 19T^{2} \) |
| 23 | \( 1 - 8.93T + 23T^{2} \) |
| 29 | \( 1 - 8.29T + 29T^{2} \) |
| 31 | \( 1 + 0.636T + 31T^{2} \) |
| 37 | \( 1 + 8.79T + 37T^{2} \) |
| 41 | \( 1 - 2.64T + 41T^{2} \) |
| 43 | \( 1 + 4.11T + 43T^{2} \) |
| 47 | \( 1 - 5.85T + 47T^{2} \) |
| 53 | \( 1 + 0.386T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 1.90T + 67T^{2} \) |
| 71 | \( 1 + 6.54T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 5.74T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 - 3.28T + 89T^{2} \) |
| 97 | \( 1 - 8.13T + 97T^{2} \) |
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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
−7.54206425388690695074677673910, −7.19689832466308333187616710781, −6.67160816594211285531848646802, −5.12667244615011134214512761731, −4.74257511994529138306252408082, −3.98610782553436027431617089317, −3.10294330714698978018236593196, −2.70536414285389877708427207264, −1.13103521182360741632742126327, 0,
1.13103521182360741632742126327, 2.70536414285389877708427207264, 3.10294330714698978018236593196, 3.98610782553436027431617089317, 4.74257511994529138306252408082, 5.12667244615011134214512761731, 6.67160816594211285531848646802, 7.19689832466308333187616710781, 7.54206425388690695074677673910
