| L(s) = 1 | + 0.713·2-s + 3.20·3-s − 1.49·4-s − 2·5-s + 2.28·6-s − 2.49·8-s + 7.26·9-s − 1.42·10-s − 2·11-s − 4.77·12-s − 5.49·13-s − 6.40·15-s + 1.20·16-s − 5.69·17-s + 5.18·18-s − 4.71·19-s + 2.98·20-s − 1.42·22-s − 0.795·23-s − 7.98·24-s − 25-s − 3.91·26-s + 13.6·27-s + 2.57·29-s − 4.57·30-s + 1.42·31-s + 5.84·32-s + ⋯ |
| L(s) = 1 | + 0.504·2-s + 1.85·3-s − 0.745·4-s − 0.894·5-s + 0.933·6-s − 0.880·8-s + 2.42·9-s − 0.451·10-s − 0.603·11-s − 1.37·12-s − 1.52·13-s − 1.65·15-s + 0.301·16-s − 1.38·17-s + 1.22·18-s − 1.08·19-s + 0.666·20-s − 0.304·22-s − 0.165·23-s − 1.62·24-s − 0.200·25-s − 0.768·26-s + 2.63·27-s + 0.477·29-s − 0.834·30-s + 0.256·31-s + 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 0.713T + 2T^{2} \) |
| 3 | \( 1 - 3.20T + 3T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 5.49T + 13T^{2} \) |
| 17 | \( 1 + 5.69T + 17T^{2} \) |
| 19 | \( 1 + 4.71T + 19T^{2} \) |
| 23 | \( 1 + 0.795T + 23T^{2} \) |
| 29 | \( 1 - 2.57T + 29T^{2} \) |
| 31 | \( 1 - 1.42T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 + 5.89T + 47T^{2} \) |
| 53 | \( 1 + 3.55T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 5.55T + 61T^{2} \) |
| 67 | \( 1 - 7.55T + 67T^{2} \) |
| 71 | \( 1 - 1.83T + 71T^{2} \) |
| 73 | \( 1 + 9.39T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 0.854T + 83T^{2} \) |
| 89 | \( 1 + 7.19T + 89T^{2} \) |
| 97 | \( 1 + 5.36T + 97T^{2} \) |
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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
−8.588016461335493521050656151630, −8.202405041730449973946826595242, −7.43126675549857140350527137595, −6.66745847174153822443668159451, −5.10779803960385738453534432341, −4.34833196089490003545012580519, −3.88765320249711006521781359838, −2.85123229080014921608369648573, −2.18472405961480177973500692731, 0,
2.18472405961480177973500692731, 2.85123229080014921608369648573, 3.88765320249711006521781359838, 4.34833196089490003545012580519, 5.10779803960385738453534432341, 6.66745847174153822443668159451, 7.43126675549857140350527137595, 8.202405041730449973946826595242, 8.588016461335493521050656151630
