L(s) = 1 | + (−0.707 + 0.707i)3-s + (0.707 + 0.707i)5-s − i·9-s + (0.707 + 0.707i)11-s + (0.707 − 0.707i)13-s − 15-s + 17-s + (0.707 − 0.707i)19-s − i·23-s + i·25-s + (0.707 + 0.707i)27-s + (0.707 − 0.707i)29-s − 31-s − 33-s + (−0.707 − 0.707i)37-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)3-s + (0.707 + 0.707i)5-s − i·9-s + (0.707 + 0.707i)11-s + (0.707 − 0.707i)13-s − 15-s + 17-s + (0.707 − 0.707i)19-s − i·23-s + i·25-s + (0.707 + 0.707i)27-s + (0.707 − 0.707i)29-s − 31-s − 33-s + (−0.707 − 0.707i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.687922890 + 0.9022135813i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.687922890 + 0.9022135813i\) |
\(L(1)\) |
\(\approx\) |
\(1.090653061 + 0.3423136927i\) |
\(L(1)\) |
\(\approx\) |
\(1.090653061 + 0.3423136927i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
| 11 | \( 1 + (0.707 + 0.707i)T \) |
| 13 | \( 1 + (0.707 - 0.707i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + (0.707 - 0.707i)T \) |
| 23 | \( 1 - iT \) |
| 29 | \( 1 + (0.707 - 0.707i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (-0.707 - 0.707i)T \) |
| 41 | \( 1 - iT \) |
| 43 | \( 1 + (0.707 + 0.707i)T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (0.707 + 0.707i)T \) |
| 59 | \( 1 + (0.707 + 0.707i)T \) |
| 61 | \( 1 + (-0.707 + 0.707i)T \) |
| 67 | \( 1 + (0.707 - 0.707i)T \) |
| 71 | \( 1 + iT \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + (0.707 - 0.707i)T \) |
| 89 | \( 1 - iT \) |
| 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
−25.7086997470811618188178244793, −25.11587078019996653718728643630, −24.10896060964159892305776667398, −23.54430433372540526104298840998, −22.317757982817606376366154149730, −21.52503582733543381799015383043, −20.53926817429235575475356655144, −19.29116898734253543975824024162, −18.50254690770067047044976065318, −17.4967050010898594330120088529, −16.66521068275058744172135320407, −16.0658624618889852045331733589, −14.10726863634340645879057237276, −13.683011735915425334182390621013, −12.43498661026730675003155714836, −11.76165226113779201127198273834, −10.597042158464399075601508853268, −9.35144838589790291580498874134, −8.32685185947896981852459829304, −7.030469073080084462774895536713, −5.91161495991977525985819205879, −5.277755541488717307565487393408, −3.65506076335518427848451754793, −1.75139763427535185238706849017, −0.95299030507813112415699471745,
1.05288190967685616543022269819, 2.837847846619779736004870369970, 4.05519217915864634290231738239, 5.38538541223822912809115688426, 6.22455425236326142676177184142, 7.28821206932803402186483241693, 9.001057924516890082460591485072, 9.94864375145313827549521650171, 10.66657247304937255467507501974, 11.6930621055830241596114974573, 12.77599374436755463275234724791, 14.13511569675605439343991055785, 14.932339345573583710069607292251, 15.925110603777765645186614585885, 17.007888430545665019520233811, 17.79816896181893701862380561872, 18.509779870710357730713370725276, 20.00742357834492830816332466257, 20.95677732953237638368512096881, 21.791918527475954285533867737906, 22.707191619157735283179954850349, 23.11961477822167127214061709131, 24.61324622149710807793476073165, 25.6005508799565157627557449415, 26.36183903316331298877948284254