| L(s) = 1 | + (116. − 37.9i)2-s + (−222. − 161. i)3-s + (8.91e3 − 6.47e3i)4-s + (698. − 2.14e3i)5-s + (−3.22e4 − 1.04e4i)6-s + (9.65e3 + 1.32e4i)7-s + (5.00e5 − 6.88e5i)8-s + (−1.40e5 − 4.33e5i)9-s − 2.77e5i·10-s + (8.56e5 + 1.55e6i)11-s − 3.03e6·12-s + (−3.90e6 + 1.26e6i)13-s + (1.63e6 + 1.18e6i)14-s + (−5.03e5 + 3.65e5i)15-s + (1.83e7 − 5.65e7i)16-s + (3.80e7 + 1.23e7i)17-s + ⋯ |
| L(s) = 1 | + (1.82 − 0.593i)2-s + (−0.305 − 0.222i)3-s + (2.17 − 1.58i)4-s + (0.0446 − 0.137i)5-s + (−0.690 − 0.224i)6-s + (0.0820 + 0.112i)7-s + (1.90 − 2.62i)8-s + (−0.264 − 0.815i)9-s − 0.277i·10-s + (0.483 + 0.875i)11-s − 1.01·12-s + (−0.808 + 0.262i)13-s + (0.216 + 0.157i)14-s + (−0.0442 + 0.0321i)15-s + (1.09 − 3.37i)16-s + (1.57 + 0.512i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.210 + 0.977i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.210 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(3.49762 - 2.82433i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.49762 - 2.82433i\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-8.56e5 - 1.55e6i)T \) |
| good | 2 | \( 1 + (-116. + 37.9i)T + (3.31e3 - 2.40e3i)T^{2} \) |
| 3 | \( 1 + (222. + 161. i)T + (1.64e5 + 5.05e5i)T^{2} \) |
| 5 | \( 1 + (-698. + 2.14e3i)T + (-1.97e8 - 1.43e8i)T^{2} \) |
| 7 | \( 1 + (-9.65e3 - 1.32e4i)T + (-4.27e9 + 1.31e10i)T^{2} \) |
| 13 | \( 1 + (3.90e6 - 1.26e6i)T + (1.88e13 - 1.36e13i)T^{2} \) |
| 17 | \( 1 + (-3.80e7 - 1.23e7i)T + (4.71e14 + 3.42e14i)T^{2} \) |
| 19 | \( 1 + (3.13e7 - 4.32e7i)T + (-6.83e14 - 2.10e15i)T^{2} \) |
| 23 | \( 1 + 7.06e7T + 2.19e16T^{2} \) |
| 29 | \( 1 + (5.75e7 + 7.92e7i)T + (-1.09e17 + 3.36e17i)T^{2} \) |
| 31 | \( 1 + (-2.87e8 - 8.86e8i)T + (-6.37e17 + 4.62e17i)T^{2} \) |
| 37 | \( 1 + (3.20e9 - 2.32e9i)T + (2.03e18 - 6.26e18i)T^{2} \) |
| 41 | \( 1 + (-1.52e9 + 2.09e9i)T + (-6.97e18 - 2.14e19i)T^{2} \) |
| 43 | \( 1 + 9.88e9iT - 3.99e19T^{2} \) |
| 47 | \( 1 + (4.66e9 + 3.38e9i)T + (3.59e19 + 1.10e20i)T^{2} \) |
| 53 | \( 1 + (7.16e9 + 2.20e10i)T + (-3.97e20 + 2.88e20i)T^{2} \) |
| 59 | \( 1 + (-1.43e10 + 1.04e10i)T + (5.49e20 - 1.69e21i)T^{2} \) |
| 61 | \( 1 + (8.38e10 + 2.72e10i)T + (2.14e21 + 1.56e21i)T^{2} \) |
| 67 | \( 1 + 1.31e10T + 8.18e21T^{2} \) |
| 71 | \( 1 + (2.15e10 - 6.64e10i)T + (-1.32e22 - 9.64e21i)T^{2} \) |
| 73 | \( 1 + (-7.61e10 - 1.04e11i)T + (-7.07e21 + 2.17e22i)T^{2} \) |
| 79 | \( 1 + (-2.14e10 + 6.95e9i)T + (4.78e22 - 3.47e22i)T^{2} \) |
| 83 | \( 1 + (-3.36e11 - 1.09e11i)T + (8.64e22 + 6.28e22i)T^{2} \) |
| 89 | \( 1 + 3.55e11T + 2.46e23T^{2} \) |
| 97 | \( 1 + (-1.36e11 - 4.19e11i)T + (-5.61e23 + 4.07e23i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.92325968272957344334755718473, −15.06355180869652040932224431789, −14.28894628797760190296403122679, −12.42093749539095214132598216879, −12.05596400121138169370802769253, −10.17127249080200704332940422265, −6.75701291134186161405340544130, −5.29441761681909923126179593767, −3.61299183623096990663165014790, −1.63097609255683824542556377246,
2.85820761545238530032073820668, 4.68083854074340781937392303825, 5.97575954694521349895198903521, 7.68634087915763794592050077069, 10.95100583726591340257895136397, 12.24422433668416182036470050011, 13.74415684114513965082274566691, 14.65949464878203224103828516810, 16.17892349674566497225144707069, 16.99685703969230482368844708106
