| L(s) = 1 | + (−0.500 − 2.83i)2-s + (−6.92 − 13.9i)3-s + (22.2 − 8.10i)4-s + (12.2 + 10.2i)5-s + (−36.1 + 26.6i)6-s + (−228. − 83.2i)7-s + (−80.2 − 138. i)8-s + (−147. + 193. i)9-s + (23.0 − 39.9i)10-s + (437. − 367. i)11-s + (−267. − 254. i)12-s + (−85.3 + 484. i)13-s + (−121. + 690. i)14-s + (58.8 − 242. i)15-s + (226. − 190. i)16-s + (435. − 753. i)17-s + ⋯ |
| L(s) = 1 | + (−0.0884 − 0.501i)2-s + (−0.444 − 0.895i)3-s + (0.695 − 0.253i)4-s + (0.219 + 0.184i)5-s + (−0.410 + 0.302i)6-s + (−1.76 − 0.641i)7-s + (−0.443 − 0.767i)8-s + (−0.605 + 0.796i)9-s + (0.0729 − 0.126i)10-s + (1.09 − 0.915i)11-s + (−0.536 − 0.510i)12-s + (−0.140 + 0.794i)13-s + (−0.165 + 0.941i)14-s + (0.0674 − 0.278i)15-s + (0.221 − 0.185i)16-s + (0.365 − 0.632i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 + 0.536i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.843 + 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.323503 - 1.11101i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.323503 - 1.11101i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (6.92 + 13.9i)T \) |
| good | 2 | \( 1 + (0.500 + 2.83i)T + (-30.0 + 10.9i)T^{2} \) |
| 5 | \( 1 + (-12.2 - 10.2i)T + (542. + 3.07e3i)T^{2} \) |
| 7 | \( 1 + (228. + 83.2i)T + (1.28e4 + 1.08e4i)T^{2} \) |
| 11 | \( 1 + (-437. + 367. i)T + (2.79e4 - 1.58e5i)T^{2} \) |
| 13 | \( 1 + (85.3 - 484. i)T + (-3.48e5 - 1.26e5i)T^{2} \) |
| 17 | \( 1 + (-435. + 753. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (33.7 + 58.4i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-4.11e3 + 1.49e3i)T + (4.93e6 - 4.13e6i)T^{2} \) |
| 29 | \( 1 + (-715. - 4.05e3i)T + (-1.92e7 + 7.01e6i)T^{2} \) |
| 31 | \( 1 + (-1.61e3 + 587. i)T + (2.19e7 - 1.84e7i)T^{2} \) |
| 37 | \( 1 + (-3.11e3 + 5.39e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + (1.38e3 - 7.84e3i)T + (-1.08e8 - 3.96e7i)T^{2} \) |
| 43 | \( 1 + (1.06e4 - 8.93e3i)T + (2.55e7 - 1.44e8i)T^{2} \) |
| 47 | \( 1 + (-854. - 311. i)T + (1.75e8 + 1.47e8i)T^{2} \) |
| 53 | \( 1 + 2.25e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + (1.30e4 + 1.09e4i)T + (1.24e8 + 7.04e8i)T^{2} \) |
| 61 | \( 1 + (-9.77e3 - 3.55e3i)T + (6.46e8 + 5.42e8i)T^{2} \) |
| 67 | \( 1 + (-432. + 2.45e3i)T + (-1.26e9 - 4.61e8i)T^{2} \) |
| 71 | \( 1 + (-1.21e4 + 2.11e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (1.30e4 + 2.26e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-5.71e3 - 3.24e4i)T + (-2.89e9 + 1.05e9i)T^{2} \) |
| 83 | \( 1 + (-4.89e3 - 2.77e4i)T + (-3.70e9 + 1.34e9i)T^{2} \) |
| 89 | \( 1 + (-5.49e3 - 9.51e3i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + (8.74e4 - 7.33e4i)T + (1.49e9 - 8.45e9i)T^{2} \) |
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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
−16.30908315880557660262912171981, −14.24618649751593492001187056074, −13.04277401739622953648681430725, −11.92538353846342018093162567350, −10.77792054141125700749564105958, −9.392247066965664823464536510018, −6.86404545992437108523655990428, −6.35762686238675641882921451696, −3.03015555152686300974977006107, −0.833473160794127886603367210553,
3.27407035012343611360303626163, 5.69017688963810823922873626115, 6.80164421843840853026716674188, 9.047341837874295633870341410690, 10.07936803245784497836431774786, 11.76984370482808973747632592716, 12.79006138197911987756496220241, 15.08945769192935696685239737968, 15.50262631273198536664812194363, 16.81663863515251770112927794979
