| L(s) = 1 | + (−12.9 − 10.8i)2-s + (−46.7 − 0.939i)3-s + (27.1 + 153. i)4-s + (355. + 129. i)5-s + (593. + 518. i)6-s + (58.7 − 333. i)7-s + (237. − 411. i)8-s + (2.18e3 + 87.8i)9-s + (−3.18e3 − 5.51e3i)10-s + (−3.88e3 + 1.41e3i)11-s + (−1.12e3 − 7.21e3i)12-s + (1.94e3 − 1.62e3i)13-s + (−4.37e3 + 3.66e3i)14-s + (−1.64e4 − 6.37e3i)15-s + (1.12e4 − 4.09e3i)16-s + (−2.67e3 − 4.62e3i)17-s + ⋯ |
| L(s) = 1 | + (−1.14 − 0.957i)2-s + (−0.999 − 0.0200i)3-s + (0.211 + 1.20i)4-s + (1.27 + 0.462i)5-s + (1.12 + 0.980i)6-s + (0.0647 − 0.367i)7-s + (0.164 − 0.284i)8-s + (0.999 + 0.0401i)9-s + (−1.00 − 1.74i)10-s + (−0.878 + 0.319i)11-s + (−0.187 − 1.20i)12-s + (0.245 − 0.205i)13-s + (−0.425 + 0.357i)14-s + (−1.26 − 0.487i)15-s + (0.687 − 0.250i)16-s + (−0.131 − 0.228i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.288 + 0.957i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.288 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.430957 - 0.580142i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.430957 - 0.580142i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (46.7 + 0.939i)T \) |
| good | 2 | \( 1 + (12.9 + 10.8i)T + (22.2 + 126. i)T^{2} \) |
| 5 | \( 1 + (-355. - 129. i)T + (5.98e4 + 5.02e4i)T^{2} \) |
| 7 | \( 1 + (-58.7 + 333. i)T + (-7.73e5 - 2.81e5i)T^{2} \) |
| 11 | \( 1 + (3.88e3 - 1.41e3i)T + (1.49e7 - 1.25e7i)T^{2} \) |
| 13 | \( 1 + (-1.94e3 + 1.62e3i)T + (1.08e7 - 6.17e7i)T^{2} \) |
| 17 | \( 1 + (2.67e3 + 4.62e3i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-2.81e4 + 4.87e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (4.59e3 + 2.60e4i)T + (-3.19e9 + 1.16e9i)T^{2} \) |
| 29 | \( 1 + (-1.61e5 - 1.35e5i)T + (2.99e9 + 1.69e10i)T^{2} \) |
| 31 | \( 1 + (1.38e4 + 7.84e4i)T + (-2.58e10 + 9.40e9i)T^{2} \) |
| 37 | \( 1 + (3.03e5 + 5.24e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-3.23e5 + 2.71e5i)T + (3.38e10 - 1.91e11i)T^{2} \) |
| 43 | \( 1 + (-4.22e5 + 1.53e5i)T + (2.08e11 - 1.74e11i)T^{2} \) |
| 47 | \( 1 + (-8.36e4 + 4.74e5i)T + (-4.76e11 - 1.73e11i)T^{2} \) |
| 53 | \( 1 - 7.48e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (2.16e6 + 7.88e5i)T + (1.90e12 + 1.59e12i)T^{2} \) |
| 61 | \( 1 + (2.28e5 - 1.29e6i)T + (-2.95e12 - 1.07e12i)T^{2} \) |
| 67 | \( 1 + (-2.85e6 + 2.39e6i)T + (1.05e12 - 5.96e12i)T^{2} \) |
| 71 | \( 1 + (-2.40e6 - 4.17e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-1.79e5 + 3.11e5i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (4.95e5 + 4.15e5i)T + (3.33e12 + 1.89e13i)T^{2} \) |
| 83 | \( 1 + (-2.10e6 - 1.76e6i)T + (4.71e12 + 2.67e13i)T^{2} \) |
| 89 | \( 1 + (-4.32e6 + 7.48e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (1.10e7 - 4.02e6i)T + (6.18e13 - 5.19e13i)T^{2} \) |
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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
−15.86362437617423971169618305186, −13.81450145025631308043626098169, −12.47744676521180219339531169413, −10.94178387233644006373473733800, −10.39634841362582568519038450501, −9.265119534116126855221373704038, −7.16139441338577117247298912986, −5.40216945888633982907664021796, −2.38211483950342988854317685305, −0.69900286759592880731670120645,
1.25510071263616923729668510200, 5.44933178696710520280608505067, 6.27883216641447664014859113298, 8.020242389479574338054759831011, 9.543355592296314353438850255571, 10.39055589576341991606430376252, 12.27679556973345431343766041044, 13.71417712099295863166717159020, 15.57971652546640370788276079648, 16.42583748255170093622289788994
