| L(s) = 1 | + (−5.50 + 1.28i)2-s + (−13.2 − 8.26i)3-s + (28.6 − 14.1i)4-s + (38.3 − 22.1i)5-s + (83.4 + 28.5i)6-s + (−102. − 59.4i)7-s + (−139. + 114. i)8-s + (106. + 218. i)9-s + (−182. + 171. i)10-s + (−349. + 604. i)11-s + (−496. − 49.8i)12-s + (−271. − 470. i)13-s + (643. + 194. i)14-s + (−690. − 24.4i)15-s + (622. − 813. i)16-s + 1.88e3i·17-s + ⋯ |
| L(s) = 1 | + (−0.973 + 0.227i)2-s + (−0.847 − 0.530i)3-s + (0.896 − 0.442i)4-s + (0.686 − 0.396i)5-s + (0.946 + 0.323i)6-s + (−0.794 − 0.458i)7-s + (−0.772 + 0.635i)8-s + (0.437 + 0.899i)9-s + (−0.578 + 0.541i)10-s + (−0.870 + 1.50i)11-s + (−0.994 − 0.0999i)12-s + (−0.446 − 0.772i)13-s + (0.877 + 0.265i)14-s + (−0.791 − 0.0280i)15-s + (0.607 − 0.794i)16-s + 1.58i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.672 - 0.740i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0787901 + 0.177926i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0787901 + 0.177926i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (5.50 - 1.28i)T \) |
| 3 | \( 1 + (13.2 + 8.26i)T \) |
| good | 5 | \( 1 + (-38.3 + 22.1i)T + (1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (102. + 59.4i)T + (8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (349. - 604. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (271. + 470. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 - 1.88e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 961. iT - 2.47e6T^{2} \) |
| 23 | \( 1 + (566. + 980. i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-173. - 99.9i)T + (1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (-225. + 130. i)T + (1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + 1.11e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (8.44e3 - 4.87e3i)T + (5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-5.30e3 - 3.06e3i)T + (7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (589. - 1.02e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + 3.40e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + (-5.31e3 - 9.21e3i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (5.47e3 - 9.47e3i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-4.97e4 + 2.87e4i)T + (6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + 7.09e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.85e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (-1.28e4 - 7.42e3i)T + (1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-1.20e3 + 2.07e3i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + 2.87e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (7.87e4 - 1.36e5i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−16.28309184126017936471599221503, −15.05853326484440741833759595382, −13.08073612611575703541481949942, −12.33791435779153622257546295848, −10.42198758425667633923961374408, −9.963854303675899634338013764909, −7.963009801780072478654024099955, −6.72866219809845098062381331987, −5.42923434882830078946227539613, −1.80700806245526016335091908093,
0.16106176997110621781787188513, 2.89363344438497030530625459796, 5.68631464350044771370713173959, 6.89792537080726119988309445467, 9.026626261448685469489435094642, 9.958181454246932853118988524124, 11.05375104191480740422157416100, 12.09484797599302800424963158784, 13.72299677842389885014698591781, 15.71619787095877137752086480908
