| L(s) = 1 | + (2.60 + 0.724i)2-s + (4.53 + 2.73i)4-s + (1.88 − 0.0730i)5-s + (−3.79 − 0.594i)7-s + (6.10 + 6.46i)8-s + (4.94 + 1.17i)10-s + (0.256 + 1.87i)11-s + (3.67 − 5.35i)13-s + (−9.45 − 4.29i)14-s + (6.25 + 11.8i)16-s + (−3.18 + 1.59i)17-s + (−0.0127 − 0.219i)19-s + (8.72 + 4.81i)20-s + (−0.693 + 5.07i)22-s + (−1.10 + 2.85i)23-s + ⋯ |
| L(s) = 1 | + (1.83 + 0.512i)2-s + (2.26 + 1.36i)4-s + (0.841 − 0.0326i)5-s + (−1.43 − 0.224i)7-s + (2.15 + 2.28i)8-s + (1.56 + 0.370i)10-s + (0.0774 + 0.566i)11-s + (1.01 − 1.48i)13-s + (−2.52 − 1.14i)14-s + (1.56 + 2.96i)16-s + (−0.771 + 0.387i)17-s + (−0.00293 − 0.0503i)19-s + (1.95 + 1.07i)20-s + (−0.147 + 1.08i)22-s + (−0.229 + 0.595i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.12260 + 1.95570i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.12260 + 1.95570i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-2.60 - 0.724i)T + (1.71 + 1.03i)T^{2} \) |
| 5 | \( 1 + (-1.88 + 0.0730i)T + (4.98 - 0.387i)T^{2} \) |
| 7 | \( 1 + (3.79 + 0.594i)T + (6.66 + 2.13i)T^{2} \) |
| 11 | \( 1 + (-0.256 - 1.87i)T + (-10.5 + 2.94i)T^{2} \) |
| 13 | \( 1 + (-3.67 + 5.35i)T + (-4.68 - 12.1i)T^{2} \) |
| 17 | \( 1 + (3.18 - 1.59i)T + (10.1 - 13.6i)T^{2} \) |
| 19 | \( 1 + (0.0127 + 0.219i)T + (-18.8 + 2.20i)T^{2} \) |
| 23 | \( 1 + (1.10 - 2.85i)T + (-17.0 - 15.4i)T^{2} \) |
| 29 | \( 1 + (0.913 + 0.653i)T + (9.38 + 27.4i)T^{2} \) |
| 31 | \( 1 + (-0.433 + 0.496i)T + (-4.19 - 30.7i)T^{2} \) |
| 37 | \( 1 + (3.46 + 4.65i)T + (-10.6 + 35.4i)T^{2} \) |
| 41 | \( 1 + (-0.698 + 2.71i)T + (-35.8 - 19.8i)T^{2} \) |
| 43 | \( 1 + (8.61 + 7.81i)T + (4.16 + 42.7i)T^{2} \) |
| 47 | \( 1 + (-3.40 - 3.90i)T + (-6.36 + 46.5i)T^{2} \) |
| 53 | \( 1 + (-1.02 + 5.78i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-6.66 + 2.72i)T + (42.1 - 41.3i)T^{2} \) |
| 61 | \( 1 + (-3.41 + 2.06i)T + (28.4 - 53.9i)T^{2} \) |
| 67 | \( 1 + (6.03 - 4.31i)T + (21.6 - 63.3i)T^{2} \) |
| 71 | \( 1 + (-3.34 - 11.1i)T + (-59.3 + 39.0i)T^{2} \) |
| 73 | \( 1 + (11.2 - 2.66i)T + (65.2 - 32.7i)T^{2} \) |
| 79 | \( 1 + (0.372 - 0.365i)T + (1.53 - 78.9i)T^{2} \) |
| 83 | \( 1 + (0.659 + 2.55i)T + (-72.6 + 40.0i)T^{2} \) |
| 89 | \( 1 + (-0.664 + 2.21i)T + (-74.3 - 48.9i)T^{2} \) |
| 97 | \( 1 + (-17.6 - 0.686i)T + (96.7 + 7.51i)T^{2} \) |
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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
−10.59607845918873810606513501910, −9.935832880688565763638980367684, −8.621579862564835314022708886664, −7.40947368414979489245733857921, −6.61997226358844329595723352227, −5.91319766265316201757545167638, −5.36361270150981297272341350458, −3.96072827095948512810738108926, −3.31813605274450831152102087664, −2.15765374046743815398647218525,
1.74632478044532658719240660243, 2.83071170788779425716726616255, 3.70795764879052552739989370985, 4.67143090152077163145202365945, 5.91774104882333861504783337375, 6.32860679037236027628170088128, 6.90005868706713785355686981696, 8.833476775047442903348407865249, 9.710450163750438963191610041414, 10.49536742191405656976467984684
