| L(s) = 1 | − 206. i·3-s − 1.39e3·5-s + 1.52e3i·7-s + 1.64e4·9-s − 2.94e5i·11-s − 1.98e5·13-s + 2.88e5i·15-s + 3.67e5·17-s + 9.19e5i·19-s + 3.15e5·21-s − 2.57e6i·23-s + 1.95e6·25-s − 1.55e7i·27-s + 1.40e7·29-s − 4.90e7i·31-s + ⋯ |
| L(s) = 1 | − 0.849i·3-s − 0.447·5-s + 0.0908i·7-s + 0.278·9-s − 1.82i·11-s − 0.533·13-s + 0.379i·15-s + 0.259·17-s + 0.371i·19-s + 0.0771·21-s − 0.400i·23-s + 0.200·25-s − 1.08i·27-s + 0.686·29-s − 1.71i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(1.650170343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.650170343\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 1.39e3T \) |
| good | 3 | \( 1 + 206. iT - 5.90e4T^{2} \) |
| 7 | \( 1 - 1.52e3iT - 2.82e8T^{2} \) |
| 11 | \( 1 + 2.94e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 + 1.98e5T + 1.37e11T^{2} \) |
| 17 | \( 1 - 3.67e5T + 2.01e12T^{2} \) |
| 19 | \( 1 - 9.19e5iT - 6.13e12T^{2} \) |
| 23 | \( 1 + 2.57e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 - 1.40e7T + 4.20e14T^{2} \) |
| 31 | \( 1 + 4.90e7iT - 8.19e14T^{2} \) |
| 37 | \( 1 - 1.21e8T + 4.80e15T^{2} \) |
| 41 | \( 1 + 1.48e8T + 1.34e16T^{2} \) |
| 43 | \( 1 - 1.41e8iT - 2.16e16T^{2} \) |
| 47 | \( 1 + 3.99e7iT - 5.25e16T^{2} \) |
| 53 | \( 1 + 1.03e8T + 1.74e17T^{2} \) |
| 59 | \( 1 - 4.38e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 - 2.83e8T + 7.13e17T^{2} \) |
| 67 | \( 1 + 2.00e9iT - 1.82e18T^{2} \) |
| 71 | \( 1 + 1.54e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 3.44e9T + 4.29e18T^{2} \) |
| 79 | \( 1 + 2.88e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 + 3.41e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 + 2.47e9T + 3.11e19T^{2} \) |
| 97 | \( 1 + 2.70e9T + 7.37e19T^{2} \) |
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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
−9.363567402845948743609862837273, −8.178758067520467644754310812291, −7.75634306706940912707963587432, −6.54535247442954319744243388099, −5.84340815815117005496716708897, −4.48702046495195020177768463533, −3.34788021856252720991722453478, −2.30471428413957611590568213672, −0.987250344852275111573504732314, −0.35864366545543602859634674520,
1.18206625053966884765761118923, 2.41470957735339896138782035271, 3.68926644035856057278198933630, 4.57023093202551840745113908411, 5.14962332969057042488081699888, 6.85486380712241787155762975709, 7.40300554045692841758574268388, 8.638746542663406596977379385377, 9.787336117749519584969648616172, 10.06920885593579486543054002413
