L(s) = 1 | + (0.766 + 0.642i)3-s + (0.939 − 0.342i)7-s + (0.173 + 0.984i)9-s + (0.5 + 0.866i)11-s + (0.173 − 0.984i)13-s + (−0.173 − 0.984i)17-s + (−0.766 − 0.642i)19-s + (0.939 + 0.342i)21-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)27-s + (0.5 + 0.866i)29-s + 31-s + (−0.173 + 0.984i)33-s + (0.766 − 0.642i)39-s + (0.173 − 0.984i)41-s + ⋯ |
L(s) = 1 | + (0.766 + 0.642i)3-s + (0.939 − 0.342i)7-s + (0.173 + 0.984i)9-s + (0.5 + 0.866i)11-s + (0.173 − 0.984i)13-s + (−0.173 − 0.984i)17-s + (−0.766 − 0.642i)19-s + (0.939 + 0.342i)21-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)27-s + (0.5 + 0.866i)29-s + 31-s + (−0.173 + 0.984i)33-s + (0.766 − 0.642i)39-s + (0.173 − 0.984i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.476790110 + 0.3633418102i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.476790110 + 0.3633418102i\) |
\(L(1)\) |
\(\approx\) |
\(1.575430788 + 0.2165614673i\) |
\(L(1)\) |
\(\approx\) |
\(1.575430788 + 0.2165614673i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 3 | \( 1 + (0.766 + 0.642i)T \) |
| 7 | \( 1 + (0.939 - 0.342i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.173 - 0.984i)T \) |
| 17 | \( 1 + (-0.173 - 0.984i)T \) |
| 19 | \( 1 + (-0.766 - 0.642i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + T \) |
| 41 | \( 1 + (0.173 - 0.984i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.5 - 0.866i)T \) |
| 53 | \( 1 + (-0.939 - 0.342i)T \) |
| 59 | \( 1 + (0.939 + 0.342i)T \) |
| 61 | \( 1 + (-0.173 + 0.984i)T \) |
| 67 | \( 1 + (-0.939 + 0.342i)T \) |
| 71 | \( 1 + (0.766 + 0.642i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + (-0.939 + 0.342i)T \) |
| 83 | \( 1 + (0.173 + 0.984i)T \) |
| 89 | \( 1 + (-0.939 - 0.342i)T \) |
| 97 | \( 1 + (0.5 - 0.866i)T \) |
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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
−20.78104791131012701932160941075, −19.64056995432095245700201878379, −19.07064013856768168911045396987, −18.70303221354965486919439635199, −17.49755623112197557233056427166, −17.21122273247521208102757564657, −15.99247527427402833137314554199, −15.141505200548636118908180181739, −14.440182084894573920527642598972, −13.93499678111865939227546299347, −13.14593148661367082447212029759, −12.21561233676461951617895148051, −11.55969892699090813774638622581, −10.814183498468298785113000770030, −9.587182781606735520321846290721, −8.81539842662337541077840228774, −8.25309009997029687892604562021, −7.58788321236595042291034227890, −6.37689949233969688310920194668, −6.01194985116746780914059863244, −4.542174793281134884004464199019, −3.85791374104474960426779437402, −2.794429012465289910284693950240, −1.815123149084908773974832214028, −1.19291056394942342350515541388,
1.01408561337545733037458515624, 2.23390625490032518044403929693, 2.91590665775762060075667840992, 4.147733021029146495672853514769, 4.64100341696936448557374010035, 5.42668597036781175556967069218, 6.879975750981367194507105513882, 7.47775363680237303488501441490, 8.49945619529081060130092359883, 8.91136364732647401291393496887, 10.01813497452499278643703496913, 10.57914232546387537386128440712, 11.3402808865877768101089405970, 12.37929362349083112228115519556, 13.238165818684584916286279622185, 14.08293138578760826383922895387, 14.65845851436376168561950661776, 15.31959841806202894136765100291, 15.980517691075087286054849215087, 17.03876794930419814371036926017, 17.61382267440683771962383269914, 18.41300664330981084140629544875, 19.462415876939581884165372040, 20.071574889171137166295125985484, 20.7471109717978366527779463742