| L(s) = 1 | + (−1.91 + 0.588i)2-s + (−1.68 − 2.48i)3-s + (3.30 − 2.24i)4-s + (−0.0158 − 0.0897i)5-s + (4.68 + 3.74i)6-s + (4.80 + 4.03i)7-s + (−5.00 + 6.24i)8-s + (−3.30 + 8.37i)9-s + (0.0830 + 0.162i)10-s + (−1.16 + 6.63i)11-s + (−11.1 − 4.41i)12-s + (3.44 − 9.45i)13-s + (−11.5 − 4.88i)14-s + (−0.196 + 0.190i)15-s + (5.89 − 14.8i)16-s + (8.06 − 4.65i)17-s + ⋯ |
| L(s) = 1 | + (−0.955 + 0.294i)2-s + (−0.562 − 0.826i)3-s + (0.827 − 0.562i)4-s + (−0.00316 − 0.0179i)5-s + (0.780 + 0.624i)6-s + (0.686 + 0.576i)7-s + (−0.625 + 0.780i)8-s + (−0.367 + 0.930i)9-s + (0.00830 + 0.0162i)10-s + (−0.106 + 0.603i)11-s + (−0.929 − 0.367i)12-s + (0.264 − 0.727i)13-s + (−0.825 − 0.348i)14-s + (−0.0130 + 0.0127i)15-s + (0.368 − 0.929i)16-s + (0.474 − 0.273i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.162i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.898864 - 0.0735135i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.898864 - 0.0735135i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.91 - 0.588i)T \) |
| 3 | \( 1 + (1.68 + 2.48i)T \) |
| good | 5 | \( 1 + (0.0158 + 0.0897i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-4.80 - 4.03i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (1.16 - 6.63i)T + (-113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-3.44 + 9.45i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-8.06 + 4.65i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-16.7 - 9.69i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-0.477 - 0.569i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-5.36 + 1.95i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (8.84 - 7.42i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (-45.0 + 26.0i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-16.1 + 44.3i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-33.2 - 5.85i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (12.2 - 14.6i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 - 31.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-10.4 - 59.4i)T + (-3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (38.9 - 46.3i)T + (-646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-25.7 + 70.8i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-22.9 + 13.2i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (11.2 - 19.4i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (136. - 49.8i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (30.3 - 11.0i)T + (5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-95.4 - 55.1i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (16.2 - 92.2i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−11.95420708305323036975656774452, −11.07014592543294918713834033304, −10.18727623535797015463511300136, −8.921728661019722002529203585489, −7.903579210690618471408627635153, −7.27214424559122420601730790408, −5.95941442195859450867561529077, −5.17553345594734183937081875627, −2.49833457575678870006508481146, −1.06618792041083124146365082513,
1.02374146405231642084592891223, 3.20010112986043174572417105654, 4.52990505614305321409404900145, 5.98993679815539971674222566119, 7.19354500275018452366090602996, 8.364313202673277995890736169888, 9.329657277723442823676335978187, 10.19789121982968561434676200071, 11.27255628481863042719161244778, 11.40036629388798651061322676613
