L(s) = 1 | + (−0.955 + 0.294i)2-s + (0.826 − 0.563i)4-s + (−0.988 − 0.149i)5-s + (−0.623 + 0.781i)8-s + (0.988 − 0.149i)10-s + (0.733 + 0.680i)11-s + (0.222 − 0.974i)13-s + (0.365 − 0.930i)16-s + (0.0747 − 0.997i)17-s + (0.5 + 0.866i)19-s + (−0.900 + 0.433i)20-s + (−0.900 − 0.433i)22-s + (−0.0747 − 0.997i)23-s + (0.955 + 0.294i)25-s + (0.0747 + 0.997i)26-s + ⋯ |
L(s) = 1 | + (−0.955 + 0.294i)2-s + (0.826 − 0.563i)4-s + (−0.988 − 0.149i)5-s + (−0.623 + 0.781i)8-s + (0.988 − 0.149i)10-s + (0.733 + 0.680i)11-s + (0.222 − 0.974i)13-s + (0.365 − 0.930i)16-s + (0.0747 − 0.997i)17-s + (0.5 + 0.866i)19-s + (−0.900 + 0.433i)20-s + (−0.900 − 0.433i)22-s + (−0.0747 − 0.997i)23-s + (0.955 + 0.294i)25-s + (0.0747 + 0.997i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6250444034 - 0.07719782382i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6250444034 - 0.07719782382i\) |
\(L(1)\) |
\(\approx\) |
\(0.6552868890 + 0.0009362437754i\) |
\(L(1)\) |
\(\approx\) |
\(0.6552868890 + 0.0009362437754i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.955 + 0.294i)T \) |
| 5 | \( 1 + (-0.988 - 0.149i)T \) |
| 11 | \( 1 + (0.733 + 0.680i)T \) |
| 13 | \( 1 + (0.222 - 0.974i)T \) |
| 17 | \( 1 + (0.0747 - 0.997i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.0747 - 0.997i)T \) |
| 29 | \( 1 + (0.900 - 0.433i)T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.826 + 0.563i)T \) |
| 41 | \( 1 + (0.623 - 0.781i)T \) |
| 43 | \( 1 + (0.623 + 0.781i)T \) |
| 47 | \( 1 + (0.955 - 0.294i)T \) |
| 53 | \( 1 + (-0.826 + 0.563i)T \) |
| 59 | \( 1 + (-0.988 + 0.149i)T \) |
| 61 | \( 1 + (-0.826 - 0.563i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.900 + 0.433i)T \) |
| 73 | \( 1 + (-0.955 - 0.294i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 + (-0.222 - 0.974i)T \) |
| 89 | \( 1 + (-0.733 + 0.680i)T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−28.13926534446631907622176821806, −27.172515589079867601211803057907, −26.545772763894717950015486767861, −25.54676585311772250689339401022, −24.30485167810207031878722876658, −23.54999644457151434499181838205, −22.0071939792379622262353650607, −21.24116044956404921607997645443, −19.6548888680024260122316428270, −19.53565775949467285037782111689, −18.36977482769106405468311511456, −17.2120860939200907188946549894, −16.22668023672981476184052017852, −15.42958897451449410154894477896, −14.06291916643086568097971037765, −12.44325857761025363957453602883, −11.517320842769418112902378193625, −10.82225489165289237703008909990, −9.31767899475512979250887062048, −8.46450658374618659091809185254, −7.33517944920092291418922859258, −6.29907634036503508575571752740, −4.16248026633437961815498503993, −3.06219033330936147764369548291, −1.24921177535168384842669961887,
0.933619498113316701513380037826, 2.83591344808341412822941986809, 4.49470973843552403020916699381, 6.07223297071615687757020675000, 7.36622195083247284699307676917, 8.115523554539010566436685334733, 9.31099777023602157169333632618, 10.41017620853175416293938825011, 11.60648395510431179029541264144, 12.405114931784344989100526658, 14.267837122534313743324318634798, 15.29354695262048661489943615459, 16.08018140171176998600694535543, 17.086282952151063001550605098832, 18.1896594646744135069519049078, 19.06304361471190519887724071558, 20.190735897709280242219010795724, 20.58458397124500475068754824820, 22.56838729755449426632251807824, 23.244628767147287698009182332964, 24.59673573124088339830962417284, 25.09430818580623156011943389818, 26.381751404750168399255618148057, 27.310913395141092741979223169965, 27.80214809187897961224510007464