| L(s) = 1 | + (−4.48 − 1.63i)2-s + (−0.520 − 2.95i)3-s + (11.3 + 9.51i)4-s + (13.4 − 11.3i)5-s + (−2.48 + 14.1i)6-s + (2.15 − 3.74i)7-s + (−16.2 − 28.1i)8-s + (−8.45 + 3.07i)9-s + (−78.9 + 28.7i)10-s + (−31.5 − 54.6i)11-s + (22.2 − 38.4i)12-s + (−8.63 + 48.9i)13-s + (−15.7 + 13.2i)14-s + (−40.4 − 33.9i)15-s + (6.38 + 36.1i)16-s + (−25.1 − 9.17i)17-s + ⋯ |
| L(s) = 1 | + (−1.58 − 0.577i)2-s + (−0.100 − 0.568i)3-s + (1.41 + 1.18i)4-s + (1.20 − 1.01i)5-s + (−0.169 + 0.959i)6-s + (0.116 − 0.201i)7-s + (−0.718 − 1.24i)8-s + (−0.313 + 0.114i)9-s + (−2.49 + 0.908i)10-s + (−0.865 − 1.49i)11-s + (0.534 − 0.925i)12-s + (−0.184 + 1.04i)13-s + (−0.301 + 0.253i)14-s + (−0.696 − 0.584i)15-s + (0.0997 + 0.565i)16-s + (−0.359 − 0.130i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 + 0.550i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.834 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.188727 - 0.628665i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.188727 - 0.628665i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.520 + 2.95i)T \) |
| 19 | \( 1 + (-6.27 + 82.5i)T \) |
| good | 2 | \( 1 + (4.48 + 1.63i)T + (6.12 + 5.14i)T^{2} \) |
| 5 | \( 1 + (-13.4 + 11.3i)T + (21.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (-2.15 + 3.74i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (31.5 + 54.6i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (8.63 - 48.9i)T + (-2.06e3 - 751. i)T^{2} \) |
| 17 | \( 1 + (25.1 + 9.17i)T + (3.76e3 + 3.15e3i)T^{2} \) |
| 23 | \( 1 + (-74.2 - 62.3i)T + (2.11e3 + 1.19e4i)T^{2} \) |
| 29 | \( 1 + (221. - 80.6i)T + (1.86e4 - 1.56e4i)T^{2} \) |
| 31 | \( 1 + (-51.9 + 89.8i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 89.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + (69.2 + 392. i)T + (-6.47e4 + 2.35e4i)T^{2} \) |
| 43 | \( 1 + (-225. + 189. i)T + (1.38e4 - 7.82e4i)T^{2} \) |
| 47 | \( 1 + (-13.5 + 4.92i)T + (7.95e4 - 6.67e4i)T^{2} \) |
| 53 | \( 1 + (-322. - 270. i)T + (2.58e4 + 1.46e5i)T^{2} \) |
| 59 | \( 1 + (-494. - 180. i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-430. - 360. i)T + (3.94e4 + 2.23e5i)T^{2} \) |
| 67 | \( 1 + (-217. + 79.2i)T + (2.30e5 - 1.93e5i)T^{2} \) |
| 71 | \( 1 + (59.8 - 50.2i)T + (6.21e4 - 3.52e5i)T^{2} \) |
| 73 | \( 1 + (-7.02 - 39.8i)T + (-3.65e5 + 1.33e5i)T^{2} \) |
| 79 | \( 1 + (-7.77 - 44.1i)T + (-4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (-380. + 659. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-255. + 1.45e3i)T + (-6.62e5 - 2.41e5i)T^{2} \) |
| 97 | \( 1 + (-141. - 51.4i)T + (6.99e5 + 5.86e5i)T^{2} \) |
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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
−13.79394986804236786749334498663, −13.08477135046865091442171776633, −11.60088906080037341204741547373, −10.67232032932825942993400678665, −9.227400038109133756153821212381, −8.743441889397642448119423832295, −7.26570736819201166203032440689, −5.54730695969568656718625481773, −2.27219744070516697770639897065, −0.78442239562833819043470716599,
2.28237410848860397678703912301, 5.52464247541677997349082314434, 6.81925752430826217420529360626, 8.037735551865910916026054978272, 9.654328912220212506384670778066, 10.08645485409700482807598923317, 10.93827845175214712494125193837, 12.97147101980297845927859759441, 14.81221714166228047733139504167, 15.15082027506612055210209364332
