| L(s) = 1 | + 3-s + 2·7-s + 9-s + 3·11-s + 13-s − 2·17-s + 8·19-s + 2·21-s + 3·23-s + 27-s − 8·29-s + 9·31-s + 3·33-s + 10·37-s + 39-s − 3·41-s − 43-s − 7·47-s − 3·49-s − 2·51-s + 5·53-s + 8·57-s + 9·59-s − 2·61-s + 2·63-s + 8·67-s + 3·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s − 0.485·17-s + 1.83·19-s + 0.436·21-s + 0.625·23-s + 0.192·27-s − 1.48·29-s + 1.61·31-s + 0.522·33-s + 1.64·37-s + 0.160·39-s − 0.468·41-s − 0.152·43-s − 1.02·47-s − 3/7·49-s − 0.280·51-s + 0.686·53-s + 1.05·57-s + 1.17·59-s − 0.256·61-s + 0.251·63-s + 0.977·67-s + 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.347993844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.347993844\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−13.15652831651306, −12.71452645711224, −11.82949762246901, −11.62987455895145, −11.34732723808007, −10.80430265638029, −9.990208066905693, −9.709952826367479, −9.321366952739167, −8.705535788115890, −8.331936859102086, −7.864170708731227, −7.272111162010710, −6.980328059019973, −6.282030547790628, −5.803440191379666, −5.100334801202349, −4.751929097639201, −4.072218955080694, −3.667702704120267, −2.991367467642648, −2.540450661222718, −1.687052927671333, −1.288548618912979, −0.6644188332442347,
0.6644188332442347, 1.288548618912979, 1.687052927671333, 2.540450661222718, 2.991367467642648, 3.667702704120267, 4.072218955080694, 4.751929097639201, 5.100334801202349, 5.803440191379666, 6.282030547790628, 6.980328059019973, 7.272111162010710, 7.864170708731227, 8.331936859102086, 8.705535788115890, 9.321366952739167, 9.709952826367479, 9.990208066905693, 10.80430265638029, 11.34732723808007, 11.62987455895145, 11.82949762246901, 12.71452645711224, 13.15652831651306
