| L(s) = 1 | + 3-s + 9-s − 4·11-s + 2·13-s + 6·17-s + 2·19-s + 2·23-s + 27-s − 2·29-s − 2·31-s − 4·33-s − 10·37-s + 2·39-s − 6·41-s + 8·43-s + 8·47-s + 6·51-s + 4·53-s + 2·57-s − 8·59-s − 8·67-s + 2·69-s − 12·71-s − 10·73-s − 8·79-s + 81-s + 4·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 1.45·17-s + 0.458·19-s + 0.417·23-s + 0.192·27-s − 0.371·29-s − 0.359·31-s − 0.696·33-s − 1.64·37-s + 0.320·39-s − 0.937·41-s + 1.21·43-s + 1.16·47-s + 0.840·51-s + 0.549·53-s + 0.264·57-s − 1.04·59-s − 0.977·67-s + 0.240·69-s − 1.42·71-s − 1.17·73-s − 0.900·79-s + 1/9·81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.46089482624679, −14.16056520267430, −13.49126481656555, −13.21929795411506, −12.64941971915152, −11.99510746211287, −11.76752685904790, −10.71650353660036, −10.53198304017278, −10.12477877163796, −9.287072182840409, −8.987876992285699, −8.345957406911317, −7.785192792877110, −7.395227979895559, −6.950409990768189, −5.935183647987903, −5.608422646765215, −5.046973952621985, −4.335181962616248, −3.549979368105229, −3.160566846030746, −2.560832065062637, −1.707035212571128, −1.071404422384804, 0,
1.071404422384804, 1.707035212571128, 2.560832065062637, 3.160566846030746, 3.549979368105229, 4.335181962616248, 5.046973952621985, 5.608422646765215, 5.935183647987903, 6.950409990768189, 7.395227979895559, 7.785192792877110, 8.345957406911317, 8.987876992285699, 9.287072182840409, 10.12477877163796, 10.53198304017278, 10.71650353660036, 11.76752685904790, 11.99510746211287, 12.64941971915152, 13.21929795411506, 13.49126481656555, 14.16056520267430, 14.46089482624679
