L(s) = 1 | + (−0.809 − 0.587i)3-s + (−0.309 + 0.951i)5-s + (−0.809 + 0.587i)7-s + (0.309 + 0.951i)9-s + (0.309 + 0.951i)13-s + (0.809 − 0.587i)15-s + (−0.309 + 0.951i)17-s + (0.809 + 0.587i)19-s + 21-s − 23-s + (−0.809 − 0.587i)25-s + (0.309 − 0.951i)27-s + (−0.809 + 0.587i)29-s + (−0.309 − 0.951i)31-s + (−0.309 − 0.951i)35-s + ⋯ |
L(s) = 1 | + (−0.809 − 0.587i)3-s + (−0.309 + 0.951i)5-s + (−0.809 + 0.587i)7-s + (0.309 + 0.951i)9-s + (0.309 + 0.951i)13-s + (0.809 − 0.587i)15-s + (−0.309 + 0.951i)17-s + (0.809 + 0.587i)19-s + 21-s − 23-s + (−0.809 − 0.587i)25-s + (0.309 − 0.951i)27-s + (−0.809 + 0.587i)29-s + (−0.309 − 0.951i)31-s + (−0.309 − 0.951i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3993263170 + 0.3724822098i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3993263170 + 0.3724822098i\) |
\(L(1)\) |
\(\approx\) |
\(0.6511998565 + 0.1691030041i\) |
\(L(1)\) |
\(\approx\) |
\(0.6511998565 + 0.1691030041i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 + (-0.309 + 0.951i)T \) |
| 7 | \( 1 + (-0.809 + 0.587i)T \) |
| 13 | \( 1 + (0.309 + 0.951i)T \) |
| 17 | \( 1 + (-0.309 + 0.951i)T \) |
| 19 | \( 1 + (0.809 + 0.587i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (-0.809 + 0.587i)T \) |
| 31 | \( 1 + (-0.309 - 0.951i)T \) |
| 37 | \( 1 + (0.809 - 0.587i)T \) |
| 41 | \( 1 + (0.809 + 0.587i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (0.809 + 0.587i)T \) |
| 53 | \( 1 + (-0.309 - 0.951i)T \) |
| 59 | \( 1 + (-0.809 + 0.587i)T \) |
| 61 | \( 1 + (0.309 - 0.951i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (-0.309 + 0.951i)T \) |
| 73 | \( 1 + (0.809 - 0.587i)T \) |
| 79 | \( 1 + (0.309 + 0.951i)T \) |
| 83 | \( 1 + (-0.309 + 0.951i)T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + (0.309 + 0.951i)T \) |
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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
−30.10478521302104464781812904979, −29.03949374369125982664707195113, −28.28250985061461518310673051557, −27.31559330771876736912893266293, −26.36022141471776852090606573934, −24.95126818769146770010766919196, −23.76190095950076553155583017055, −22.88687695821551701342466745430, −21.984685361679824037337430526045, −20.51275472771878567823296017407, −19.97948368370428560071277490483, −18.2186459006013946919096410136, −17.10454449068520966488626544543, −16.13624430958668194312003575203, −15.533727943879889723251744949625, −13.58206509377565419165009291057, −12.51669672961695940352484854460, −11.40979936600001759350941875620, −10.105392733237627371851516793084, −9.12338134511641725793708558653, −7.43718119626603726998764887568, −5.91354156194166474795720552202, −4.74351485585035498836217597940, −3.490087663771641315866403262917, −0.64820396794601972718801604188,
2.10579865845431469113989746090, 3.8325979303635796474161533732, 5.837641127410206019420611613822, 6.61033101181637062184098660776, 7.85754244879477082874470058159, 9.65545083914649846185624880254, 10.97502698410018377562435467139, 11.89742657093131609595517938831, 13.01391950443119377870755231272, 14.33958443889979078147039412861, 15.73503815556105305394697873528, 16.68470150962378205705302441197, 18.182297020305848063206113976867, 18.74639336295990101897218168864, 19.74478703322054946154751739927, 21.72037190153563865092499952300, 22.34838352224467146167472126830, 23.33299980424584101588597259494, 24.30690786395969530056456566479, 25.63296929619169314578889525074, 26.51980993386904844611656024126, 27.95035823637056982002728213219, 28.77019452360659402392553364447, 29.73840506480344196746090531603, 30.70893128527336636085071586713