| L(s) = 1 | + 3-s + 5-s − 2·9-s + 4·11-s − 2·13-s + 15-s − 9·23-s + 25-s − 5·27-s − 2·29-s + 4·33-s − 9·37-s − 2·39-s + 2·41-s − 8·43-s − 2·45-s + 6·47-s + 2·53-s + 4·55-s + 3·59-s − 4·61-s − 2·65-s − 2·67-s − 9·69-s − 4·71-s + 73-s + 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 2/3·9-s + 1.20·11-s − 0.554·13-s + 0.258·15-s − 1.87·23-s + 1/5·25-s − 0.962·27-s − 0.371·29-s + 0.696·33-s − 1.47·37-s − 0.320·39-s + 0.312·41-s − 1.21·43-s − 0.298·45-s + 0.875·47-s + 0.274·53-s + 0.539·55-s + 0.390·59-s − 0.512·61-s − 0.248·65-s − 0.244·67-s − 1.08·69-s − 0.474·71-s + 0.117·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.578248989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.578248989\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 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
−12.67418509878231, −12.16981166663176, −11.88400982661008, −11.51761818320350, −10.88856725341042, −10.27658121894938, −9.979791155801811, −9.459965494543062, −8.985498927061271, −8.669539855225837, −8.206773551417802, −7.545990992807903, −7.271522798349479, −6.414388187986243, −6.289000890340696, −5.622684649497143, −5.191247893441667, −4.539472721603984, −3.805357676053298, −3.675402056963972, −2.906843104436739, −2.311529018549318, −1.862662955874472, −1.323593208385543, −0.3049459053209243,
0.3049459053209243, 1.323593208385543, 1.862662955874472, 2.311529018549318, 2.906843104436739, 3.675402056963972, 3.805357676053298, 4.539472721603984, 5.191247893441667, 5.622684649497143, 6.289000890340696, 6.414388187986243, 7.271522798349479, 7.545990992807903, 8.206773551417802, 8.669539855225837, 8.985498927061271, 9.459965494543062, 9.979791155801811, 10.27658121894938, 10.88856725341042, 11.51761818320350, 11.88400982661008, 12.16981166663176, 12.67418509878231
