| L(s) = 1 | + 3·3-s + 5-s − 7-s + 6·9-s − 5·11-s + 3·13-s + 3·15-s + 17-s − 6·19-s − 3·21-s − 6·23-s + 25-s + 9·27-s + 9·29-s + 4·31-s − 15·33-s − 35-s + 9·39-s − 4·41-s − 10·43-s + 6·45-s − 47-s + 49-s + 3·51-s + 4·53-s − 5·55-s − 18·57-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s − 0.377·7-s + 2·9-s − 1.50·11-s + 0.832·13-s + 0.774·15-s + 0.242·17-s − 1.37·19-s − 0.654·21-s − 1.25·23-s + 1/5·25-s + 1.73·27-s + 1.67·29-s + 0.718·31-s − 2.61·33-s − 0.169·35-s + 1.44·39-s − 0.624·41-s − 1.52·43-s + 0.894·45-s − 0.145·47-s + 1/7·49-s + 0.420·51-s + 0.549·53-s − 0.674·55-s − 2.38·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 191660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−13.35430214192104, −13.06245026630998, −12.69284862242375, −12.09103534052503, −11.48341307244280, −10.70219036609037, −10.30208505155534, −9.925637592115784, −9.769090311337749, −8.800694713626529, −8.545076603778132, −8.129137886372791, −7.995628076338837, −7.102260928885852, −6.625024881505416, −6.262840432216659, −5.462268818438806, −4.976990285017213, −4.249664752085837, −3.821636006662730, −3.258365737341771, −2.618925731698685, −2.376213646157021, −1.783744515696667, −1.002760081241508, 0,
1.002760081241508, 1.783744515696667, 2.376213646157021, 2.618925731698685, 3.258365737341771, 3.821636006662730, 4.249664752085837, 4.976990285017213, 5.462268818438806, 6.262840432216659, 6.625024881505416, 7.102260928885852, 7.995628076338837, 8.129137886372791, 8.545076603778132, 8.800694713626529, 9.769090311337749, 9.925637592115784, 10.30208505155534, 10.70219036609037, 11.48341307244280, 12.09103534052503, 12.69284862242375, 13.06245026630998, 13.35430214192104
