| L(s) = 1 | + 29.2·3-s − 25·5-s + 44.5·7-s + 611.·9-s − 349.·11-s + 255.·13-s − 730.·15-s + 1.50e3·17-s + 2.43e3·19-s + 1.30e3·21-s + 1.43e3·23-s + 625·25-s + 1.07e4·27-s + 2.87e3·29-s − 8.94e3·31-s − 1.02e4·33-s − 1.11e3·35-s − 1.45e4·37-s + 7.45e3·39-s + 7.50e3·41-s + 1.34e4·43-s − 1.52e4·45-s + 8.44e3·47-s − 1.48e4·49-s + 4.39e4·51-s − 2.83e4·53-s + 8.73e3·55-s + ⋯ |
| L(s) = 1 | + 1.87·3-s − 0.447·5-s + 0.343·7-s + 2.51·9-s − 0.870·11-s + 0.418·13-s − 0.838·15-s + 1.26·17-s + 1.54·19-s + 0.643·21-s + 0.565·23-s + 0.200·25-s + 2.84·27-s + 0.634·29-s − 1.67·31-s − 1.63·33-s − 0.153·35-s − 1.74·37-s + 0.784·39-s + 0.697·41-s + 1.11·43-s − 1.12·45-s + 0.557·47-s − 0.882·49-s + 2.36·51-s − 1.38·53-s + 0.389·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.853676218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.853676218\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| good | 3 | \( 1 - 29.2T + 243T^{2} \) |
| 7 | \( 1 - 44.5T + 1.68e4T^{2} \) |
| 11 | \( 1 + 349.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 255.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.50e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.43e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.43e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.87e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.94e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.45e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.50e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.34e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 8.44e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.83e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.53e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.56e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.98e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.00e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.92e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 981.T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.83e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.94e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.13e4T + 8.58e9T^{2} \) |
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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
−12.25563175392576554413584563725, −10.77595980459570363843061478455, −9.704948015342368619048192252063, −8.792722971364665246147493874558, −7.81977183620813337860427884549, −7.32489583251520492713429655407, −5.19288498165303801902364402961, −3.69003441553887275972559230985, −2.87925918490213238237340748663, −1.36419258460414701214751002118,
1.36419258460414701214751002118, 2.87925918490213238237340748663, 3.69003441553887275972559230985, 5.19288498165303801902364402961, 7.32489583251520492713429655407, 7.81977183620813337860427884549, 8.792722971364665246147493874558, 9.704948015342368619048192252063, 10.77595980459570363843061478455, 12.25563175392576554413584563725
