| 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 | \( 1 + (-1.36e11 + 4.19e11i)T + (-5.61e23 - 4.07e23i)T^{2} \) |
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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
−16.99685703969230482368844708106, −16.17892349674566497225144707069, −14.65949464878203224103828516810, −13.74415684114513965082274566691, −12.24422433668416182036470050011, −10.95100583726591340257895136397, −7.68634087915763794592050077069, −5.97575954694521349895198903521, −4.68083854074340781937392303825, −2.85820761545238530032073820668,
1.63097609255683824542556377246, 3.61299183623096990663165014790, 5.29441761681909923126179593767, 6.75701291134186161405340544130, 10.17127249080200704332940422265, 12.05596400121138169370802769253, 12.42093749539095214132598216879, 14.28894628797760190296403122679, 15.06355180869652040932224431789, 16.92325968272957344334755718473
