| L(s) = 1 | + (5.65 + 9.79i)2-s + (126. + 73.0i)3-s + (−63.9 + 110. i)4-s + (372. − 215. i)5-s + 1.65e3i·6-s + (−1.54e3 − 1.83e3i)7-s − 1.44e3·8-s + (7.37e3 + 1.27e4i)9-s + (4.21e3 + 2.43e3i)10-s + (−5.18e3 + 8.97e3i)11-s + (−1.61e4 + 9.34e3i)12-s − 2.42e4i·13-s + (9.25e3 − 2.55e4i)14-s + 6.28e4·15-s + (−8.19e3 − 1.41e4i)16-s + (−9.15e4 − 5.28e4i)17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (1.56 + 0.901i)3-s + (−0.249 + 0.433i)4-s + (0.596 − 0.344i)5-s + 1.27i·6-s + (−0.643 − 0.765i)7-s − 0.353·8-s + (1.12 + 1.94i)9-s + (0.421 + 0.243i)10-s + (−0.354 + 0.613i)11-s + (−0.780 + 0.450i)12-s − 0.847i·13-s + (0.241 − 0.664i)14-s + 1.24·15-s + (−0.125 − 0.216i)16-s + (−1.09 − 0.632i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.21794 + 1.77477i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.21794 + 1.77477i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-5.65 - 9.79i)T \) |
| 7 | \( 1 + (1.54e3 + 1.83e3i)T \) |
| good | 3 | \( 1 + (-126. - 73.0i)T + (3.28e3 + 5.68e3i)T^{2} \) |
| 5 | \( 1 + (-372. + 215. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 11 | \( 1 + (5.18e3 - 8.97e3i)T + (-1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + 2.42e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (9.15e4 + 5.28e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-1.85e5 + 1.06e5i)T + (8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (1.67e4 + 2.90e4i)T + (-3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 - 7.45e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + (-2.71e5 - 1.56e5i)T + (4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (1.10e6 + 1.92e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 - 2.59e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 4.49e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-1.61e6 + 9.34e5i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (3.31e6 - 5.73e6i)T + (-3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-4.50e6 - 2.59e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (2.00e7 - 1.15e7i)T + (9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (5.75e6 - 9.97e6i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 - 2.59e5T + 6.45e14T^{2} \) |
| 73 | \( 1 + (7.31e6 + 4.22e6i)T + (4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (2.85e5 + 4.95e5i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + 6.46e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-1.52e7 + 8.80e6i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 - 7.31e7iT - 7.83e15T^{2} \) |
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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
−17.76090733714863648143429719861, −16.10858192317741273008572326739, −15.32766333792234081172072741177, −13.83969782497069320972827971119, −13.23139781753217261077872795564, −10.11333398184574034515772753061, −9.063700154968992151578751343602, −7.42209378316194747176878460628, −4.74810937183079476484236899639, −3.00590690460346104072954771131,
1.93637335508627641129831712451, 3.19550590132415873808056945169, 6.43933200814760964041412529792, 8.527153423346676740521559141171, 9.776831083653402360505291837953, 12.12181381579597250187160093829, 13.46028933841217581370731583929, 14.10191020359451295418735947141, 15.55433916444375833884743474302, 18.17489850645827183431526809560
