| L(s) = 1 | + (−9.32 − 3.39i)2-s + (39.8 + 24.5i)3-s + (−22.5 − 18.9i)4-s + (12.2 − 69.5i)5-s + (−287. − 364. i)6-s + (48.0 − 40.3i)7-s + (781. + 1.35e3i)8-s + (981. + 1.95e3i)9-s + (−350. + 607. i)10-s + (−1.38e3 − 7.83e3i)11-s + (−433. − 1.30e3i)12-s + (1.28e4 − 4.68e3i)13-s + (−585. + 213. i)14-s + (2.19e3 − 2.46e3i)15-s + (−2.03e3 − 1.15e4i)16-s + (1.57e4 − 2.73e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.824 − 0.300i)2-s + (0.851 + 0.524i)3-s + (−0.176 − 0.147i)4-s + (0.0438 − 0.248i)5-s + (−0.544 − 0.688i)6-s + (0.0529 − 0.0444i)7-s + (0.539 + 0.934i)8-s + (0.448 + 0.893i)9-s + (−0.110 + 0.191i)10-s + (−0.312 − 1.77i)11-s + (−0.0723 − 0.218i)12-s + (1.62 − 0.590i)13-s + (−0.0570 + 0.0207i)14-s + (0.167 − 0.188i)15-s + (−0.124 − 0.706i)16-s + (0.779 − 1.35i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.23135 - 0.596014i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23135 - 0.596014i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-39.8 - 24.5i)T \) |
| good | 2 | \( 1 + (9.32 + 3.39i)T + (98.0 + 82.2i)T^{2} \) |
| 5 | \( 1 + (-12.2 + 69.5i)T + (-7.34e4 - 2.67e4i)T^{2} \) |
| 7 | \( 1 + (-48.0 + 40.3i)T + (1.43e5 - 8.11e5i)T^{2} \) |
| 11 | \( 1 + (1.38e3 + 7.83e3i)T + (-1.83e7 + 6.66e6i)T^{2} \) |
| 13 | \( 1 + (-1.28e4 + 4.68e3i)T + (4.80e7 - 4.03e7i)T^{2} \) |
| 17 | \( 1 + (-1.57e4 + 2.73e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-7.93e3 - 1.37e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (3.04e4 + 2.55e4i)T + (5.91e8 + 3.35e9i)T^{2} \) |
| 29 | \( 1 + (1.69e4 + 6.16e3i)T + (1.32e10 + 1.10e10i)T^{2} \) |
| 31 | \( 1 + (-8.47e3 - 7.11e3i)T + (4.77e9 + 2.70e10i)T^{2} \) |
| 37 | \( 1 + (1.82e5 - 3.15e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (4.21e5 - 1.53e5i)T + (1.49e11 - 1.25e11i)T^{2} \) |
| 43 | \( 1 + (7.23e4 + 4.10e5i)T + (-2.55e11 + 9.29e10i)T^{2} \) |
| 47 | \( 1 + (-5.01e5 + 4.20e5i)T + (8.79e10 - 4.98e11i)T^{2} \) |
| 53 | \( 1 + 2.61e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-5.85e3 + 3.31e4i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (-1.49e6 + 1.25e6i)T + (5.45e11 - 3.09e12i)T^{2} \) |
| 67 | \( 1 + (-1.23e6 + 4.50e5i)T + (4.64e12 - 3.89e12i)T^{2} \) |
| 71 | \( 1 + (1.32e6 - 2.28e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-1.00e6 - 1.73e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (4.61e6 + 1.67e6i)T + (1.47e13 + 1.23e13i)T^{2} \) |
| 83 | \( 1 + (-6.73e6 - 2.45e6i)T + (2.07e13 + 1.74e13i)T^{2} \) |
| 89 | \( 1 + (-3.51e5 - 6.08e5i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (1.89e6 + 1.07e7i)T + (-7.59e13 + 2.76e13i)T^{2} \) |
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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
−15.84053812067804838298612951581, −14.12045976255851183855168219787, −13.50412263996377889762294590940, −11.18515479192358658821981765139, −10.17405849911384212730034449147, −8.803792615263020516843888321217, −8.154820889185599952458682091759, −5.38390123651034779547506823350, −3.26408819216615068710314790860, −0.975359872648698394776087152262,
1.56958316349304482932951167193, 3.87393554275101706688231437950, 6.76938620894547533589648686027, 7.931656298701344478378847866799, 9.039135187732485191767749924202, 10.25986839327040965277669639225, 12.41933060546975208896645118195, 13.42238423770600893152037901092, 14.78720134808405475257763674309, 15.96465201201229726131180684897
