| L(s) = 1 | − 16.3·2-s + 140.·4-s − 125·5-s − 716.·7-s − 205.·8-s + 2.04e3·10-s + 1.00e3·11-s + 7.39e3·13-s + 1.17e4·14-s − 1.46e4·16-s + 2.73e4·17-s + 5.19e4·19-s − 1.75e4·20-s − 1.64e4·22-s − 8.39e4·23-s + 1.56e4·25-s − 1.21e5·26-s − 1.00e5·28-s + 2.54e5·29-s − 1.75e5·31-s + 2.65e5·32-s − 4.47e5·34-s + 8.95e4·35-s − 3.19e4·37-s − 8.51e5·38-s + 2.57e4·40-s + 2.37e5·41-s + ⋯ |
| L(s) = 1 | − 1.44·2-s + 1.09·4-s − 0.447·5-s − 0.789·7-s − 0.142·8-s + 0.647·10-s + 0.228·11-s + 0.933·13-s + 1.14·14-s − 0.892·16-s + 1.34·17-s + 1.73·19-s − 0.491·20-s − 0.330·22-s − 1.43·23-s + 0.199·25-s − 1.35·26-s − 0.867·28-s + 1.93·29-s − 1.06·31-s + 1.43·32-s − 1.95·34-s + 0.353·35-s − 0.103·37-s − 2.51·38-s + 0.0636·40-s + 0.537·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.9017925202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9017925202\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 125T \) |
| good | 2 | \( 1 + 16.3T + 128T^{2} \) |
| 7 | \( 1 + 716.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.00e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 7.39e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.73e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.19e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.39e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.54e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.75e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.19e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.37e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.44e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.81e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 5.77e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.08e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.99e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 5.83e3T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.30e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 7.54e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.68e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.75e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 7.58e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.39e7T + 8.07e13T^{2} \) |
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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
−9.913017230793388832647527839978, −9.257229543350489378750854898315, −8.260821696070457684653493887890, −7.61398564154687162748754882862, −6.66210430838449712245030359555, −5.57194193160094526720942405877, −3.97202837510814488661006699340, −2.95770281192939542372123229024, −1.37635503110904004121213014099, −0.61062775667479057955626900938,
0.61062775667479057955626900938, 1.37635503110904004121213014099, 2.95770281192939542372123229024, 3.97202837510814488661006699340, 5.57194193160094526720942405877, 6.66210430838449712245030359555, 7.61398564154687162748754882862, 8.260821696070457684653493887890, 9.257229543350489378750854898315, 9.913017230793388832647527839978
