| L(s) = 1 | + (−16.3 + 19.5i)2-s + (163. − 179. i)3-s + (65.1 + 369. i)4-s + (−95.4 + 262. i)5-s + (824. + 6.13e3i)6-s + (−1.03e3 + 5.84e3i)7-s + (−3.08e4 − 1.78e4i)8-s + (−5.46e3 − 5.87e4i)9-s + (−3.55e3 − 6.15e3i)10-s + (9.21e4 + 2.53e5i)11-s + (7.70e4 + 4.87e4i)12-s + (1.57e5 − 1.32e5i)13-s + (−9.71e4 − 1.15e5i)14-s + (3.14e4 + 6.00e4i)15-s + (4.92e5 − 1.79e5i)16-s + (1.07e5 − 6.17e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.511 + 0.609i)2-s + (0.673 − 0.739i)3-s + (0.0636 + 0.360i)4-s + (−0.0305 + 0.0839i)5-s + (0.106 + 0.788i)6-s + (−0.0612 + 0.347i)7-s + (−0.941 − 0.543i)8-s + (−0.0924 − 0.995i)9-s + (−0.0355 − 0.0615i)10-s + (0.572 + 1.57i)11-s + (0.309 + 0.195i)12-s + (0.423 − 0.355i)13-s + (−0.180 − 0.215i)14-s + (0.0414 + 0.0791i)15-s + (0.469 − 0.170i)16-s + (0.0753 − 0.0435i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.541239 + 1.15087i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.541239 + 1.15087i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-163. + 179. i)T \) |
| good | 2 | \( 1 + (16.3 - 19.5i)T + (-177. - 1.00e3i)T^{2} \) |
| 5 | \( 1 + (95.4 - 262. i)T + (-7.48e6 - 6.27e6i)T^{2} \) |
| 7 | \( 1 + (1.03e3 - 5.84e3i)T + (-2.65e8 - 9.66e7i)T^{2} \) |
| 11 | \( 1 + (-9.21e4 - 2.53e5i)T + (-1.98e10 + 1.66e10i)T^{2} \) |
| 13 | \( 1 + (-1.57e5 + 1.32e5i)T + (2.39e10 - 1.35e11i)T^{2} \) |
| 17 | \( 1 + (-1.07e5 + 6.17e4i)T + (1.00e12 - 1.74e12i)T^{2} \) |
| 19 | \( 1 + (2.05e6 - 3.56e6i)T + (-3.06e12 - 5.30e12i)T^{2} \) |
| 23 | \( 1 + (1.12e7 - 1.98e6i)T + (3.89e13 - 1.41e13i)T^{2} \) |
| 29 | \( 1 + (1.82e7 - 2.17e7i)T + (-7.30e13 - 4.14e14i)T^{2} \) |
| 31 | \( 1 + (-3.26e6 - 1.85e7i)T + (-7.70e14 + 2.80e14i)T^{2} \) |
| 37 | \( 1 + (-4.00e7 - 6.93e7i)T + (-2.40e15 + 4.16e15i)T^{2} \) |
| 41 | \( 1 + (1.03e7 + 1.23e7i)T + (-2.33e15 + 1.32e16i)T^{2} \) |
| 43 | \( 1 + (8.90e6 - 3.23e6i)T + (1.65e16 - 1.38e16i)T^{2} \) |
| 47 | \( 1 + (5.40e7 + 9.52e6i)T + (4.94e16 + 1.79e16i)T^{2} \) |
| 53 | \( 1 + 4.50e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (1.66e8 - 4.56e8i)T + (-3.91e17 - 3.28e17i)T^{2} \) |
| 61 | \( 1 + (-1.64e8 + 9.30e8i)T + (-6.70e17 - 2.43e17i)T^{2} \) |
| 67 | \( 1 + (-5.19e8 + 4.35e8i)T + (3.16e17 - 1.79e18i)T^{2} \) |
| 71 | \( 1 + (1.55e8 - 8.97e7i)T + (1.62e18 - 2.81e18i)T^{2} \) |
| 73 | \( 1 + (-1.08e9 + 1.88e9i)T + (-2.14e18 - 3.72e18i)T^{2} \) |
| 79 | \( 1 + (-1.69e9 - 1.42e9i)T + (1.64e18 + 9.32e18i)T^{2} \) |
| 83 | \( 1 + (-2.59e8 + 3.09e8i)T + (-2.69e18 - 1.52e19i)T^{2} \) |
| 89 | \( 1 + (3.94e9 + 2.27e9i)T + (1.55e19 + 2.70e19i)T^{2} \) |
| 97 | \( 1 + (-7.82e9 + 2.84e9i)T + (5.64e19 - 4.74e19i)T^{2} \) |
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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
−15.36236551648635248131449063217, −14.49855994658451783760655986503, −12.76546740076551267099304283824, −12.09796661673286857590841785393, −9.751246068893818669662772213651, −8.515232859544989614914619505139, −7.46917125677122177405776220796, −6.32412078710339127742348348067, −3.61443107242750240813886154438, −1.81010546404250860150134134213,
0.51739158898441375574694396401, 2.39962568866777335730032513917, 4.02821864328925383533507129912, 6.04118099655037103752350430720, 8.401960645104843120051938839691, 9.308370926499823192224409484220, 10.60299835936638040587476882618, 11.44518387671847237405656562742, 13.56390008308597331707803163964, 14.51674468188854219895333098334
