| L(s) = 1 | + 3-s + 5-s + 2·7-s + 9-s + 4·11-s + 15-s + 2·17-s + 8·19-s + 2·21-s + 8·23-s + 25-s + 27-s − 31-s + 4·33-s + 2·35-s + 8·37-s − 6·41-s + 45-s − 4·47-s − 3·49-s + 2·51-s + 6·53-s + 4·55-s + 8·57-s − 10·59-s − 14·61-s + 2·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.20·11-s + 0.258·15-s + 0.485·17-s + 1.83·19-s + 0.436·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.179·31-s + 0.696·33-s + 0.338·35-s + 1.31·37-s − 0.937·41-s + 0.149·45-s − 0.583·47-s − 3/7·49-s + 0.280·51-s + 0.824·53-s + 0.539·55-s + 1.05·57-s − 1.30·59-s − 1.79·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.011664298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.011664298\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.78305840154654831305354555206, −7.35052251726659103423767287367, −6.59039147011842733986375487464, −5.76744718330719275844699417356, −5.00945001444381465193937994468, −4.39517802879979283819326079938, −3.34527911522195243892086657234, −2.86868243706883710075782267847, −1.55180381446384542937234980663, −1.16938335597462296576811234455,
1.16938335597462296576811234455, 1.55180381446384542937234980663, 2.86868243706883710075782267847, 3.34527911522195243892086657234, 4.39517802879979283819326079938, 5.00945001444381465193937994468, 5.76744718330719275844699417356, 6.59039147011842733986375487464, 7.35052251726659103423767287367, 7.78305840154654831305354555206
