| L(s) = 1 | + (4.86 + 8.43i)2-s + (−34.7 − 60.2i)3-s + (16.5 − 28.6i)4-s − 459.·5-s + (338. − 586. i)6-s + (430. − 745. i)7-s + 1.56e3·8-s + (−1.32e3 + 2.29e3i)9-s + (−2.23e3 − 3.87e3i)10-s + (459. + 796. i)11-s − 2.30e3·12-s + (−7.91e3 + 383. i)13-s + 8.38e3·14-s + (1.59e4 + 2.76e4i)15-s + (5.52e3 + 9.56e3i)16-s + (4.98e3 − 8.63e3i)17-s + ⋯ |
| L(s) = 1 | + (0.430 + 0.745i)2-s + (−0.743 − 1.28i)3-s + (0.129 − 0.224i)4-s − 1.64·5-s + (0.639 − 1.10i)6-s + (0.474 − 0.821i)7-s + 1.08·8-s + (−0.605 + 1.04i)9-s + (−0.707 − 1.22i)10-s + (0.104 + 0.180i)11-s − 0.384·12-s + (−0.998 + 0.0484i)13-s + 0.816·14-s + (1.22 + 2.11i)15-s + (0.337 + 0.583i)16-s + (0.246 − 0.426i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.205 + 0.978i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.662070 - 0.815469i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.662070 - 0.815469i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (7.91e3 - 383. i)T \) |
| good | 2 | \( 1 + (-4.86 - 8.43i)T + (-64 + 110. i)T^{2} \) |
| 3 | \( 1 + (34.7 + 60.2i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + 459.T + 7.81e4T^{2} \) |
| 7 | \( 1 + (-430. + 745. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-459. - 796. i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 17 | \( 1 + (-4.98e3 + 8.63e3i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-2.82e4 + 4.90e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-9.83e3 - 1.70e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-3.10e4 - 5.37e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 - 8.90e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + (2.11e5 + 3.65e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (3.38e4 + 5.86e4i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (6.71e4 - 1.16e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + 5.12e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.34e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (5.56e5 - 9.63e5i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (7.30e5 - 1.26e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.41e6 + 2.44e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (1.60e5 - 2.78e5i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 - 1.27e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 9.11e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 9.66e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-4.32e6 - 7.49e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-5.65e6 + 9.80e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−17.73587127160628403996514034117, −16.45144668266911419234163161322, −15.16818330797627076140686937693, −13.71705680086805193486124577631, −12.09696450663236031252224143603, −11.08831539622395870750934930034, −7.48178369516946898648623468552, −7.12429956443831671175525201934, −4.84811047180588778426762025999, −0.71371123018371938950690153215,
3.56909060108583650080607022706, 4.88383535948579067049919075713, 7.994310297803712792429891575952, 10.32690049502323073963693653930, 11.74931211764021264065793969469, 12.03731356432811458587748024147, 14.85005584643824144582888820857, 15.92366967135004775144442764828, 16.86772749584885571001792052217, 19.00571383042806144719997393285
