| L(s) = 1 | + (2.78 − 0.491i)2-s + (−8.46 + 3.05i)3-s + (7.51 − 2.73i)4-s + (9.30 − 11.0i)5-s + (−22.0 + 12.6i)6-s + (82.9 + 30.1i)7-s + (19.5 − 11.3i)8-s + (62.3 − 51.6i)9-s + (20.4 − 35.4i)10-s + (67.7 + 80.7i)11-s + (−55.3 + 46.1i)12-s + (−0.524 + 2.97i)13-s + (245. + 43.3i)14-s + (−44.9 + 122. i)15-s + (49.0 − 41.1i)16-s + (−150. − 87.1i)17-s + ⋯ |
| L(s) = 1 | + (0.696 − 0.122i)2-s + (−0.940 + 0.338i)3-s + (0.469 − 0.171i)4-s + (0.372 − 0.443i)5-s + (−0.613 + 0.351i)6-s + (1.69 + 0.616i)7-s + (0.306 − 0.176i)8-s + (0.770 − 0.637i)9-s + (0.204 − 0.354i)10-s + (0.560 + 0.667i)11-s + (−0.384 + 0.320i)12-s + (−0.00310 + 0.0176i)13-s + (1.25 + 0.221i)14-s + (−0.199 + 0.543i)15-s + (0.191 − 0.160i)16-s + (−0.522 − 0.301i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0484i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.998 - 0.0484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.13733 + 0.0517853i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.13733 + 0.0517853i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.78 + 0.491i)T \) |
| 3 | \( 1 + (8.46 - 3.05i)T \) |
| good | 5 | \( 1 + (-9.30 + 11.0i)T + (-108. - 615. i)T^{2} \) |
| 7 | \( 1 + (-82.9 - 30.1i)T + (1.83e3 + 1.54e3i)T^{2} \) |
| 11 | \( 1 + (-67.7 - 80.7i)T + (-2.54e3 + 1.44e4i)T^{2} \) |
| 13 | \( 1 + (0.524 - 2.97i)T + (-2.68e4 - 9.76e3i)T^{2} \) |
| 17 | \( 1 + (150. + 87.1i)T + (4.17e4 + 7.23e4i)T^{2} \) |
| 19 | \( 1 + (100. + 174. i)T + (-6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (164. + 450. i)T + (-2.14e5 + 1.79e5i)T^{2} \) |
| 29 | \( 1 + (19.8 - 3.50i)T + (6.64e5 - 2.41e5i)T^{2} \) |
| 31 | \( 1 + (-858. + 312. i)T + (7.07e5 - 5.93e5i)T^{2} \) |
| 37 | \( 1 + (1.27e3 - 2.21e3i)T + (-9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 + (3.12e3 + 550. i)T + (2.65e6 + 9.66e5i)T^{2} \) |
| 43 | \( 1 + (745. - 625. i)T + (5.93e5 - 3.36e6i)T^{2} \) |
| 47 | \( 1 + (-986. + 2.71e3i)T + (-3.73e6 - 3.13e6i)T^{2} \) |
| 53 | \( 1 + 4.00e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (1.29e3 - 1.54e3i)T + (-2.10e6 - 1.19e7i)T^{2} \) |
| 61 | \( 1 + (-821. - 299. i)T + (1.06e7 + 8.89e6i)T^{2} \) |
| 67 | \( 1 + (848. - 4.81e3i)T + (-1.89e7 - 6.89e6i)T^{2} \) |
| 71 | \( 1 + (-3.47e3 - 2.00e3i)T + (1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 + (4.09e3 + 7.09e3i)T + (-1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (1.54e3 + 8.75e3i)T + (-3.66e7 + 1.33e7i)T^{2} \) |
| 83 | \( 1 + (8.88e3 - 1.56e3i)T + (4.45e7 - 1.62e7i)T^{2} \) |
| 89 | \( 1 + (4.22e3 - 2.43e3i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 97 | \( 1 + (2.54e3 - 2.13e3i)T + (1.53e7 - 8.71e7i)T^{2} \) |
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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
−14.78485341237266201878514316727, −13.39737485647174362244140661434, −12.01465096880033677447033240205, −11.53287842979616543502587873590, −10.22097532119752245049839552907, −8.652386184341033558979786948753, −6.75732026742390056092919818020, −5.23329068380792902843265865344, −4.55116010360477420344491897231, −1.69036823325812002161244622498,
1.61386579123354579768536209307, 4.29409982638152401198962924489, 5.61403513287565915199793406236, 6.85931078856887083703949630514, 8.147590193898817449739546994224, 10.49938859215047116248840386040, 11.22732679716365470060083129859, 12.17725190134479383493547633812, 13.71204132657457489711955207186, 14.26647276513301583371692391904
