L(s) = 1 | + (−0.149 − 0.988i)2-s + (−0.955 + 0.294i)4-s + (0.365 − 0.930i)5-s + (0.955 + 0.294i)7-s + (0.433 + 0.900i)8-s + (−0.974 − 0.222i)10-s + (0.563 + 0.826i)11-s + (−0.0747 − 0.997i)13-s + (0.149 − 0.988i)14-s + (0.826 − 0.563i)16-s + i·17-s + (0.974 + 0.222i)19-s + (−0.0747 + 0.997i)20-s + (0.733 − 0.680i)22-s + (0.988 + 0.149i)23-s + ⋯ |
L(s) = 1 | + (−0.149 − 0.988i)2-s + (−0.955 + 0.294i)4-s + (0.365 − 0.930i)5-s + (0.955 + 0.294i)7-s + (0.433 + 0.900i)8-s + (−0.974 − 0.222i)10-s + (0.563 + 0.826i)11-s + (−0.0747 − 0.997i)13-s + (0.149 − 0.988i)14-s + (0.826 − 0.563i)16-s + i·17-s + (0.974 + 0.222i)19-s + (−0.0747 + 0.997i)20-s + (0.733 − 0.680i)22-s + (0.988 + 0.149i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0681 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0681 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9186263382 - 0.8580167332i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9186263382 - 0.8580167332i\) |
\(L(1)\) |
\(\approx\) |
\(0.9374944365 - 0.5635234200i\) |
\(L(1)\) |
\(\approx\) |
\(0.9374944365 - 0.5635234200i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (-0.149 - 0.988i)T \) |
| 5 | \( 1 + (0.365 - 0.930i)T \) |
| 7 | \( 1 + (0.955 + 0.294i)T \) |
| 11 | \( 1 + (0.563 + 0.826i)T \) |
| 13 | \( 1 + (-0.0747 - 0.997i)T \) |
| 17 | \( 1 + iT \) |
| 19 | \( 1 + (0.974 + 0.222i)T \) |
| 23 | \( 1 + (0.988 + 0.149i)T \) |
| 31 | \( 1 + (-0.930 - 0.365i)T \) |
| 37 | \( 1 + (-0.433 - 0.900i)T \) |
| 41 | \( 1 + (-0.866 + 0.5i)T \) |
| 43 | \( 1 + (0.930 - 0.365i)T \) |
| 47 | \( 1 + (-0.563 - 0.826i)T \) |
| 53 | \( 1 + (-0.623 - 0.781i)T \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
| 61 | \( 1 + (0.294 - 0.955i)T \) |
| 67 | \( 1 + (-0.826 - 0.563i)T \) |
| 71 | \( 1 + (-0.900 - 0.433i)T \) |
| 73 | \( 1 + (0.781 + 0.623i)T \) |
| 79 | \( 1 + (-0.997 - 0.0747i)T \) |
| 83 | \( 1 + (0.733 + 0.680i)T \) |
| 89 | \( 1 + (-0.781 + 0.623i)T \) |
| 97 | \( 1 + (0.680 - 0.733i)T \) |
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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
−26.139277159897817653457678748813, −25.1293170906004705823393663841, −24.343076899370021860015274518132, −23.57633578376965232644053633319, −22.474091304800720645519160838766, −21.84228990205923500595386451838, −20.75334147436543929492126747910, −19.240795341683829363445398080877, −18.53235923787930325015280833095, −17.70683284963109361897063672692, −16.85787532278657128386003583478, −15.90759316541239634367853494913, −14.68610510914800875794596021284, −14.13913712488265278019635514497, −13.510901317642350410125507053698, −11.68256026305705195485133410322, −10.86183344401468530024706291004, −9.58296007610092514786606963446, −8.74145416610245313643249862948, −7.40886419628671965692351268642, −6.82603963574205968775789824759, −5.62840190508624208301074733898, −4.578146757342181885305152072364, −3.21250345554380714318748928091, −1.38018074412927636775605797659,
1.20837014936185318046496542104, 2.070144971946592612821112568366, 3.66915725637122809773470875399, 4.85417471341481574244862371879, 5.55826348167781197612314018247, 7.603189525044815034140422439616, 8.553428002922181243306189389029, 9.397996256246822635759878245893, 10.394749597873551912899215557009, 11.50519124397601287573933520080, 12.41950650712988131279852868647, 13.06850175817375204596788314473, 14.26668599506901788365047020049, 15.19601040309247937312582807238, 16.7829300068893227389107548998, 17.5541305400047868642650322399, 18.12488155151518044445034740522, 19.48233805217454505349935972977, 20.31329569758509826324811073042, 20.88224175254605627765847466954, 21.788083477847542894568529478525, 22.698958770003736280303940557412, 23.79764635803601219335856994726, 24.79236696586311125093610190440, 25.60001697968046386975511738650