| L(s) = 1 | − 2-s − 4-s + 2·5-s + 3·8-s − 2·10-s + 6·11-s + 13-s − 16-s + 17-s − 4·19-s − 2·20-s − 6·22-s − 6·23-s − 25-s − 26-s + 6·29-s + 2·31-s − 5·32-s − 34-s + 2·37-s + 4·38-s + 6·40-s − 6·41-s − 6·44-s + 6·46-s − 4·47-s + 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.06·8-s − 0.632·10-s + 1.80·11-s + 0.277·13-s − 1/4·16-s + 0.242·17-s − 0.917·19-s − 0.447·20-s − 1.27·22-s − 1.25·23-s − 1/5·25-s − 0.196·26-s + 1.11·29-s + 0.359·31-s − 0.883·32-s − 0.171·34-s + 0.328·37-s + 0.648·38-s + 0.948·40-s − 0.937·41-s − 0.904·44-s + 0.884·46-s − 0.583·47-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.95519141410564, −13.66157868951961, −13.21036596913273, −12.41465764819961, −12.18055154578874, −11.52059330519823, −10.95416714372816, −10.40672689318627, −9.899157249251742, −9.551366561261584, −9.211425215696145, −8.554426468337395, −8.195592451299666, −7.746171135640352, −6.676601685459218, −6.594110568712284, −6.035206763889542, −5.349089961825709, −4.690423319593169, −4.109249362430988, −3.762667906155529, −2.884319136471102, −1.919499257033144, −1.599546645440323, −0.9466098330543203, 0,
0.9466098330543203, 1.599546645440323, 1.919499257033144, 2.884319136471102, 3.762667906155529, 4.109249362430988, 4.690423319593169, 5.349089961825709, 6.035206763889542, 6.594110568712284, 6.676601685459218, 7.746171135640352, 8.195592451299666, 8.554426468337395, 9.211425215696145, 9.551366561261584, 9.899157249251742, 10.40672689318627, 10.95416714372816, 11.52059330519823, 12.18055154578874, 12.41465764819961, 13.21036596913273, 13.66157868951961, 13.95519141410564
