| L(s) = 1 | + (−2.50 + 0.329i)2-s + (1.66 − 0.483i)3-s + (4.21 − 1.13i)4-s + (1.60 + 3.25i)5-s + (−4.00 + 1.75i)6-s + (−1.58 − 0.779i)7-s + (−5.52 + 2.28i)8-s + (2.53 − 1.60i)9-s + (−5.09 − 7.62i)10-s + (0.00416 − 0.00141i)11-s + (6.47 − 3.91i)12-s + (0.705 + 2.63i)13-s + (4.21 + 1.42i)14-s + (4.24 + 4.64i)15-s + (5.49 − 3.17i)16-s + (−4.10 + 0.383i)17-s + ⋯ |
| L(s) = 1 | + (−1.76 + 0.232i)2-s + (0.960 − 0.278i)3-s + (2.10 − 0.565i)4-s + (0.718 + 1.45i)5-s + (−1.63 + 0.717i)6-s + (−0.597 − 0.294i)7-s + (−1.95 + 0.808i)8-s + (0.844 − 0.535i)9-s + (−1.61 − 2.41i)10-s + (0.00125 − 0.000426i)11-s + (1.86 − 1.13i)12-s + (0.195 + 0.730i)13-s + (1.12 + 0.382i)14-s + (1.09 + 1.19i)15-s + (1.37 − 0.793i)16-s + (−0.995 + 0.0931i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 - 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.757 - 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.707368 + 0.262601i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.707368 + 0.262601i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.66 + 0.483i)T \) |
| 17 | \( 1 + (4.10 - 0.383i)T \) |
| good | 2 | \( 1 + (2.50 - 0.329i)T + (1.93 - 0.517i)T^{2} \) |
| 5 | \( 1 + (-1.60 - 3.25i)T + (-3.04 + 3.96i)T^{2} \) |
| 7 | \( 1 + (1.58 + 0.779i)T + (4.26 + 5.55i)T^{2} \) |
| 11 | \( 1 + (-0.00416 + 0.00141i)T + (8.72 - 6.69i)T^{2} \) |
| 13 | \( 1 + (-0.705 - 2.63i)T + (-11.2 + 6.5i)T^{2} \) |
| 19 | \( 1 + (-6.69 - 2.77i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-3.32 + 3.79i)T + (-3.00 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.296 + 4.52i)T + (-28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (0.107 - 0.315i)T + (-24.5 - 18.8i)T^{2} \) |
| 37 | \( 1 + (-0.151 - 0.761i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (6.87 - 0.450i)T + (40.6 - 5.35i)T^{2} \) |
| 43 | \( 1 + (7.91 + 6.07i)T + (11.1 + 41.5i)T^{2} \) |
| 47 | \( 1 + (1.26 - 4.72i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (1.83 - 4.42i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-1.15 + 8.80i)T + (-56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (-3.56 + 7.22i)T + (-37.1 - 48.3i)T^{2} \) |
| 67 | \( 1 + (6.84 + 3.95i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-7.33 + 1.45i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (4.87 + 3.25i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-1.58 - 4.68i)T + (-62.6 + 48.0i)T^{2} \) |
| 83 | \( 1 + (1.07 + 8.14i)T + (-80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (-1.47 + 1.47i)T - 89iT^{2} \) |
| 97 | \( 1 + (-0.313 - 0.0205i)T + (96.1 + 12.6i)T^{2} \) |
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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
−13.38637283025129532738414742560, −11.63630945958649042735503117518, −10.52983889890939689716966186390, −9.829749354328179079789050303943, −9.142369971407135531866874222439, −7.935622452054787149945892381940, −6.84068778082507157971289642327, −6.52084857332052952153042356532, −3.19316556434545011305529585656, −1.95332552369550340010042901421,
1.39395688370224965924847989122, 2.95748971332956670756781407455, 5.18923668865278414340494073554, 6.99003564784872503678181003316, 8.207072657257264835440993379544, 8.996585345189362410280014912580, 9.440779292005487880795986147973, 10.25066227898047280854104899358, 11.61482321068789510550149139488, 12.93268141095462326758435170462
