| L(s) = 1 | + (10.7 + 3.49i)2-s + (43.1 − 31.3i)3-s + (103. + 75.2i)4-s + (−227. − 700. i)5-s + (573. − 186. i)6-s + (1.73e3 − 2.39e3i)7-s + (851. + 1.17e3i)8-s + (−1.14e3 + 3.53e3i)9-s − 8.32e3i·10-s + (−1.53e3 − 1.45e4i)11-s + 6.82e3·12-s + (2.20e4 + 7.15e3i)13-s + (2.70e4 − 1.96e4i)14-s + (−3.17e4 − 2.30e4i)15-s + (5.06e3 + 1.55e4i)16-s + (7.23e4 − 2.35e4i)17-s + ⋯ |
| L(s) = 1 | + (0.672 + 0.218i)2-s + (0.532 − 0.386i)3-s + (0.404 + 0.293i)4-s + (−0.363 − 1.12i)5-s + (0.442 − 0.143i)6-s + (0.724 − 0.997i)7-s + (0.207 + 0.286i)8-s + (−0.175 + 0.539i)9-s − 0.832i·10-s + (−0.104 − 0.994i)11-s + 0.329·12-s + (0.770 + 0.250i)13-s + (0.705 − 0.512i)14-s + (−0.626 − 0.455i)15-s + (0.0772 + 0.237i)16-s + (0.866 − 0.281i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.652 + 0.757i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.652 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.63259 - 1.20685i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.63259 - 1.20685i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-10.7 - 3.49i)T \) |
| 11 | \( 1 + (1.53e3 + 1.45e4i)T \) |
| good | 3 | \( 1 + (-43.1 + 31.3i)T + (2.02e3 - 6.23e3i)T^{2} \) |
| 5 | \( 1 + (227. + 700. i)T + (-3.16e5 + 2.29e5i)T^{2} \) |
| 7 | \( 1 + (-1.73e3 + 2.39e3i)T + (-1.78e6 - 5.48e6i)T^{2} \) |
| 13 | \( 1 + (-2.20e4 - 7.15e3i)T + (6.59e8 + 4.79e8i)T^{2} \) |
| 17 | \( 1 + (-7.23e4 + 2.35e4i)T + (5.64e9 - 4.10e9i)T^{2} \) |
| 19 | \( 1 + (-5.96e4 - 8.21e4i)T + (-5.24e9 + 1.61e10i)T^{2} \) |
| 23 | \( 1 + 4.65e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + (6.19e5 - 8.52e5i)T + (-1.54e11 - 4.75e11i)T^{2} \) |
| 31 | \( 1 + (2.80e5 - 8.63e5i)T + (-6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (-3.36e5 - 2.44e5i)T + (1.08e12 + 3.34e12i)T^{2} \) |
| 41 | \( 1 + (-1.14e6 - 1.57e6i)T + (-2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 + 2.26e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (-7.70e6 + 5.59e6i)T + (7.35e12 - 2.26e13i)T^{2} \) |
| 53 | \( 1 + (1.54e6 - 4.75e6i)T + (-5.03e13 - 3.65e13i)T^{2} \) |
| 59 | \( 1 + (3.49e6 + 2.53e6i)T + (4.53e13 + 1.39e14i)T^{2} \) |
| 61 | \( 1 + (9.43e6 - 3.06e6i)T + (1.55e14 - 1.12e14i)T^{2} \) |
| 67 | \( 1 - 2.55e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + (-7.53e6 - 2.31e7i)T + (-5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (-1.72e7 + 2.37e7i)T + (-2.49e14 - 7.66e14i)T^{2} \) |
| 79 | \( 1 + (-7.07e6 - 2.29e6i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (-9.64e6 + 3.13e6i)T + (1.82e15 - 1.32e15i)T^{2} \) |
| 89 | \( 1 + 2.35e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + (2.84e7 - 8.75e7i)T + (-6.34e15 - 4.60e15i)T^{2} \) |
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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
−16.19830226800979745332977073074, −14.17354318605584831463901965079, −13.69648017042790814280618075914, −12.30077674883959730135884666407, −10.88015915765448186264276690580, −8.503227264423072152789799320764, −7.63887570061268886437311003012, −5.38414616387730918578530982994, −3.78371282859435265000505473927, −1.30328300137933721130508084664,
2.41399291269026496306949123308, 3.88304628999322615809043478907, 5.93083576939181325243141751859, 7.81060502542929148506180199720, 9.651292695333981410437618946828, 11.19437944009924054841088954392, 12.25441461554272485143397815903, 14.11007135783039614570983453968, 15.02730042684919866342600494652, 15.53976437873526826696985454905
