| L(s) = 1 | + (−38.9 + 46.3i)2-s + (−46.9 − 238. i)3-s + (−459. − 2.60e3i)4-s + (−1.83e3 + 5.04e3i)5-s + (1.28e4 + 7.10e3i)6-s + (−3.57e3 + 2.02e4i)7-s + (8.49e4 + 4.90e4i)8-s + (−5.46e4 + 2.23e4i)9-s + (−1.62e5 − 2.81e5i)10-s + (4.88e4 + 1.34e5i)11-s + (−5.99e5 + 2.31e5i)12-s + (−1.44e5 + 1.21e5i)13-s + (−8.02e5 − 9.56e5i)14-s + (1.28e6 + 2.01e5i)15-s + (−3.03e6 + 1.10e6i)16-s + (1.12e5 − 6.51e4i)17-s + ⋯ |
| L(s) = 1 | + (−1.21 + 1.44i)2-s + (−0.193 − 0.981i)3-s + (−0.448 − 2.54i)4-s + (−0.587 + 1.61i)5-s + (1.65 + 0.913i)6-s + (−0.212 + 1.20i)7-s + (2.59 + 1.49i)8-s + (−0.925 + 0.378i)9-s + (−1.62 − 2.81i)10-s + (0.303 + 0.833i)11-s + (−2.40 + 0.930i)12-s + (−0.388 + 0.326i)13-s + (−1.49 − 1.77i)14-s + (1.69 + 0.264i)15-s + (−2.89 + 1.05i)16-s + (0.0794 − 0.0458i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.343 + 0.939i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.140168 - 0.0980286i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.140168 - 0.0980286i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (46.9 + 238. i)T \) |
| good | 2 | \( 1 + (38.9 - 46.3i)T + (-177. - 1.00e3i)T^{2} \) |
| 5 | \( 1 + (1.83e3 - 5.04e3i)T + (-7.48e6 - 6.27e6i)T^{2} \) |
| 7 | \( 1 + (3.57e3 - 2.02e4i)T + (-2.65e8 - 9.66e7i)T^{2} \) |
| 11 | \( 1 + (-4.88e4 - 1.34e5i)T + (-1.98e10 + 1.66e10i)T^{2} \) |
| 13 | \( 1 + (1.44e5 - 1.21e5i)T + (2.39e10 - 1.35e11i)T^{2} \) |
| 17 | \( 1 + (-1.12e5 + 6.51e4i)T + (1.00e12 - 1.74e12i)T^{2} \) |
| 19 | \( 1 + (3.88e5 - 6.72e5i)T + (-3.06e12 - 5.30e12i)T^{2} \) |
| 23 | \( 1 + (1.21e6 - 2.13e5i)T + (3.89e13 - 1.41e13i)T^{2} \) |
| 29 | \( 1 + (3.86e6 - 4.60e6i)T + (-7.30e13 - 4.14e14i)T^{2} \) |
| 31 | \( 1 + (-1.47e6 - 8.34e6i)T + (-7.70e14 + 2.80e14i)T^{2} \) |
| 37 | \( 1 + (6.91e7 + 1.19e8i)T + (-2.40e15 + 4.16e15i)T^{2} \) |
| 41 | \( 1 + (1.20e7 + 1.43e7i)T + (-2.33e15 + 1.32e16i)T^{2} \) |
| 43 | \( 1 + (-1.89e8 + 6.91e7i)T + (1.65e16 - 1.38e16i)T^{2} \) |
| 47 | \( 1 + (4.77e7 + 8.42e6i)T + (4.94e16 + 1.79e16i)T^{2} \) |
| 53 | \( 1 - 7.90e7iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (-2.33e8 + 6.40e8i)T + (-3.91e17 - 3.28e17i)T^{2} \) |
| 61 | \( 1 + (2.81e8 - 1.59e9i)T + (-6.70e17 - 2.43e17i)T^{2} \) |
| 67 | \( 1 + (6.50e8 - 5.46e8i)T + (3.16e17 - 1.79e18i)T^{2} \) |
| 71 | \( 1 + (1.99e9 - 1.15e9i)T + (1.62e18 - 2.81e18i)T^{2} \) |
| 73 | \( 1 + (-1.63e9 + 2.83e9i)T + (-2.14e18 - 3.72e18i)T^{2} \) |
| 79 | \( 1 + (3.03e8 + 2.55e8i)T + (1.64e18 + 9.32e18i)T^{2} \) |
| 83 | \( 1 + (5.56e7 - 6.63e7i)T + (-2.69e18 - 1.52e19i)T^{2} \) |
| 89 | \( 1 + (-5.40e9 - 3.12e9i)T + (1.55e19 + 2.70e19i)T^{2} \) |
| 97 | \( 1 + (-1.05e10 + 3.84e9i)T + (5.64e19 - 4.74e19i)T^{2} \) |
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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
−15.97555925983468281302391642505, −14.91835241025288194636211546815, −14.29491466543206695886772763640, −12.02620810500003958532454491312, −10.57854112568115089534869259240, −9.024521786114368543082149830369, −7.57848830217722170057255737844, −6.87176497472128800302170374633, −5.80188512378552149165812770684, −2.15411802880620779951339752760,
0.13152044130100720966645254807, 0.961999357568859486757894554092, 3.49099117085818549151372900301, 4.52982800940434666456787225898, 8.012678336062701164665966805945, 8.992818895718811279706217044035, 10.04175337268131392063095195390, 11.17773528013164724717976297362, 12.19342766107626877212165700533, 13.44329020657940186926722317683
