| L(s) = 1 | − 3·3-s + 5·5-s − 32·7-s + 9·9-s + 60·11-s − 34·13-s − 15·15-s + 42·17-s + 76·19-s + 96·21-s + 25·25-s − 27·27-s + 6·29-s + 232·31-s − 180·33-s − 160·35-s + 134·37-s + 102·39-s + 234·41-s + 412·43-s + 45·45-s + 360·47-s + 681·49-s − 126·51-s + 222·53-s + 300·55-s − 228·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.72·7-s + 1/3·9-s + 1.64·11-s − 0.725·13-s − 0.258·15-s + 0.599·17-s + 0.917·19-s + 0.997·21-s + 1/5·25-s − 0.192·27-s + 0.0384·29-s + 1.34·31-s − 0.949·33-s − 0.772·35-s + 0.595·37-s + 0.418·39-s + 0.891·41-s + 1.46·43-s + 0.149·45-s + 1.11·47-s + 1.98·49-s − 0.345·51-s + 0.575·53-s + 0.735·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.381458855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.381458855\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 + 32 T + p^{3} T^{2} \) |
| 11 | \( 1 - 60 T + p^{3} T^{2} \) |
| 13 | \( 1 + 34 T + p^{3} T^{2} \) |
| 17 | \( 1 - 42 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 - 6 T + p^{3} T^{2} \) |
| 31 | \( 1 - 232 T + p^{3} T^{2} \) |
| 37 | \( 1 - 134 T + p^{3} T^{2} \) |
| 41 | \( 1 - 234 T + p^{3} T^{2} \) |
| 43 | \( 1 - 412 T + p^{3} T^{2} \) |
| 47 | \( 1 - 360 T + p^{3} T^{2} \) |
| 53 | \( 1 - 222 T + p^{3} T^{2} \) |
| 59 | \( 1 + 660 T + p^{3} T^{2} \) |
| 61 | \( 1 + 490 T + p^{3} T^{2} \) |
| 67 | \( 1 + 812 T + p^{3} T^{2} \) |
| 71 | \( 1 + 120 T + p^{3} T^{2} \) |
| 73 | \( 1 - 746 T + p^{3} T^{2} \) |
| 79 | \( 1 + 152 T + p^{3} T^{2} \) |
| 83 | \( 1 - 804 T + p^{3} T^{2} \) |
| 89 | \( 1 + 678 T + p^{3} T^{2} \) |
| 97 | \( 1 - 2 p T + p^{3} T^{2} \) |
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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
−12.02151068864953700031821676826, −10.60362992023463776755777530640, −9.565705474307436268467110366903, −9.297649442040515380923893855631, −7.40410364501993588597233822616, −6.45888382928588275741405051813, −5.80253207724290559981330747927, −4.21684633378668931570811532143, −2.92557378018420706935012034758, −0.908989042364750767351828203005,
0.908989042364750767351828203005, 2.92557378018420706935012034758, 4.21684633378668931570811532143, 5.80253207724290559981330747927, 6.45888382928588275741405051813, 7.40410364501993588597233822616, 9.297649442040515380923893855631, 9.565705474307436268467110366903, 10.60362992023463776755777530640, 12.02151068864953700031821676826
