| L(s) = 1 | − 2·2-s + 2·4-s + 4·5-s − 7-s − 8·10-s − 4·13-s + 2·14-s − 4·16-s − 2·17-s − 5·19-s + 8·20-s − 9·23-s + 11·25-s + 8·26-s − 2·28-s − 29-s − 8·31-s + 8·32-s + 4·34-s − 4·35-s + 8·37-s + 10·38-s − 3·41-s + 6·43-s + 18·46-s + 7·47-s + 49-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 1.78·5-s − 0.377·7-s − 2.52·10-s − 1.10·13-s + 0.534·14-s − 16-s − 0.485·17-s − 1.14·19-s + 1.78·20-s − 1.87·23-s + 11/5·25-s + 1.56·26-s − 0.377·28-s − 0.185·29-s − 1.43·31-s + 1.41·32-s + 0.685·34-s − 0.676·35-s + 1.31·37-s + 1.62·38-s − 0.468·41-s + 0.914·43-s + 2.65·46-s + 1.02·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−13.20164772195777, −12.65431601361334, −12.42667214040607, −11.66046009279905, −11.02144151931925, −10.61039252543233, −10.28351790539056, −9.809933489340625, −9.432125820066510, −9.182227093458275, −8.717835574123531, −7.990055925928452, −7.715006348883856, −6.945609516722497, −6.679146390863761, −5.993553311967626, −5.794590914150817, −5.059538073439033, −4.414185830644186, −3.974507208597788, −2.865161931769188, −2.408143174153762, −1.917507093153010, −1.671333873962367, −0.6505149855902559, 0,
0.6505149855902559, 1.671333873962367, 1.917507093153010, 2.408143174153762, 2.865161931769188, 3.974507208597788, 4.414185830644186, 5.059538073439033, 5.794590914150817, 5.993553311967626, 6.679146390863761, 6.945609516722497, 7.715006348883856, 7.990055925928452, 8.717835574123531, 9.182227093458275, 9.432125820066510, 9.809933489340625, 10.28351790539056, 10.61039252543233, 11.02144151931925, 11.66046009279905, 12.42667214040607, 12.65431601361334, 13.20164772195777
