L(s) = 1 | − 0.190·2-s − 3-s − 1.96·4-s − 2.28·5-s + 0.190·6-s + 0.753·8-s + 9-s + 0.434·10-s + 1.72·11-s + 1.96·12-s − 1.61·13-s + 2.28·15-s + 3.78·16-s + 3.96·17-s − 0.190·18-s − 1.10·19-s + 4.48·20-s − 0.328·22-s − 2.20·23-s − 0.753·24-s + 0.213·25-s + 0.307·26-s − 27-s + 7.83·29-s − 0.434·30-s − 3.24·31-s − 2.22·32-s + ⋯ |
L(s) = 1 | − 0.134·2-s − 0.577·3-s − 0.981·4-s − 1.02·5-s + 0.0776·6-s + 0.266·8-s + 0.333·9-s + 0.137·10-s + 0.521·11-s + 0.566·12-s − 0.448·13-s + 0.589·15-s + 0.946·16-s + 0.962·17-s − 0.0448·18-s − 0.254·19-s + 1.00·20-s − 0.0701·22-s − 0.460·23-s − 0.153·24-s + 0.0427·25-s + 0.0602·26-s − 0.192·27-s + 1.45·29-s − 0.0792·30-s − 0.581·31-s − 0.393·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6278288221\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6278288221\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 + 0.190T + 2T^{2} \) |
| 5 | \( 1 + 2.28T + 5T^{2} \) |
| 11 | \( 1 - 1.72T + 11T^{2} \) |
| 13 | \( 1 + 1.61T + 13T^{2} \) |
| 17 | \( 1 - 3.96T + 17T^{2} \) |
| 19 | \( 1 + 1.10T + 19T^{2} \) |
| 23 | \( 1 + 2.20T + 23T^{2} \) |
| 29 | \( 1 - 7.83T + 29T^{2} \) |
| 31 | \( 1 + 3.24T + 31T^{2} \) |
| 37 | \( 1 - 1.90T + 37T^{2} \) |
| 43 | \( 1 - 2.91T + 43T^{2} \) |
| 47 | \( 1 + 5.86T + 47T^{2} \) |
| 53 | \( 1 + 2.97T + 53T^{2} \) |
| 59 | \( 1 + 5.61T + 59T^{2} \) |
| 61 | \( 1 + 4.86T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 - 0.406T + 73T^{2} \) |
| 79 | \( 1 + 2.23T + 79T^{2} \) |
| 83 | \( 1 - 1.25T + 83T^{2} \) |
| 89 | \( 1 + 8.84T + 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
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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
−7.932165895488492649956202424690, −7.63360301341795669126372733874, −6.65971420514771274243070148745, −5.89554202538460311987121765227, −5.06423826215688276732236500228, −4.43524439724640159749912432642, −3.84950040051976753608808939666, −3.03504558777575910235490997944, −1.49214575989798564887943244507, −0.46976463337647541018749586675,
0.46976463337647541018749586675, 1.49214575989798564887943244507, 3.03504558777575910235490997944, 3.84950040051976753608808939666, 4.43524439724640159749912432642, 5.06423826215688276732236500228, 5.89554202538460311987121765227, 6.65971420514771274243070148745, 7.63360301341795669126372733874, 7.932165895488492649956202424690