L(s) = 1 | + (−0.826 − 0.563i)2-s + (0.365 + 0.930i)4-s + (0.222 − 0.974i)5-s + (0.222 − 0.974i)8-s + (−0.733 + 0.680i)10-s + (0.900 − 0.433i)11-s + (0.826 + 0.563i)13-s + (−0.733 + 0.680i)16-s + (−0.365 + 0.930i)17-s + (−0.5 + 0.866i)19-s + (0.988 − 0.149i)20-s + (−0.988 − 0.149i)22-s + (−0.623 + 0.781i)23-s + (−0.900 − 0.433i)25-s + (−0.365 − 0.930i)26-s + ⋯ |
L(s) = 1 | + (−0.826 − 0.563i)2-s + (0.365 + 0.930i)4-s + (0.222 − 0.974i)5-s + (0.222 − 0.974i)8-s + (−0.733 + 0.680i)10-s + (0.900 − 0.433i)11-s + (0.826 + 0.563i)13-s + (−0.733 + 0.680i)16-s + (−0.365 + 0.930i)17-s + (−0.5 + 0.866i)19-s + (0.988 − 0.149i)20-s + (−0.988 − 0.149i)22-s + (−0.623 + 0.781i)23-s + (−0.900 − 0.433i)25-s + (−0.365 − 0.930i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.569 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.569 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8010193166 + 0.4197077989i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8010193166 + 0.4197077989i\) |
\(L(1)\) |
\(\approx\) |
\(0.7320757977 - 0.1318716810i\) |
\(L(1)\) |
\(\approx\) |
\(0.7320757977 - 0.1318716810i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.826 - 0.563i)T \) |
| 5 | \( 1 + (0.222 - 0.974i)T \) |
| 11 | \( 1 + (0.900 - 0.433i)T \) |
| 13 | \( 1 + (0.826 + 0.563i)T \) |
| 17 | \( 1 + (-0.365 + 0.930i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.623 + 0.781i)T \) |
| 29 | \( 1 + (0.988 - 0.149i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (-0.988 + 0.149i)T \) |
| 41 | \( 1 + (-0.955 + 0.294i)T \) |
| 43 | \( 1 + (0.955 + 0.294i)T \) |
| 47 | \( 1 + (-0.826 - 0.563i)T \) |
| 53 | \( 1 + (0.988 + 0.149i)T \) |
| 59 | \( 1 + (-0.955 - 0.294i)T \) |
| 61 | \( 1 + (0.365 - 0.930i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.623 + 0.781i)T \) |
| 73 | \( 1 + (0.0747 + 0.997i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 + (-0.826 + 0.563i)T \) |
| 89 | \( 1 + (-0.826 + 0.563i)T \) |
| 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
−23.87091066276542973674214351272, −22.76990834138080307087305719333, −22.36928005429316671112791166701, −20.98214623322358679751953265765, −20.02337765322481717805471302307, −19.28730210468366970017654033140, −18.253234700338676447220410284747, −17.85218825322210581884760409058, −16.92452399941667705701517555997, −15.82673018011329601443727773416, −15.15385821905493088158624413153, −14.280317133068038448296431632675, −13.52104193960688884337478701480, −11.945219985469376422466141513379, −11.00156791786905531584640126805, −10.31060231584060092707160055898, −9.324677270733364580991978490950, −8.47919112627890909821527363922, −7.26154840152647370615171735055, −6.63401432392856516261955461243, −5.7936383350856448673369040457, −4.41088339011434588387736176013, −2.90013843572889682598214578283, −1.793773154545198151421864088233, −0.33006567431137937163511249610,
1.230669030894256114338247948158, 1.81619276082896097062922724562, 3.54420532268699508841826120674, 4.26781311102234673165136044212, 5.86578570429871424621489040902, 6.79020045855726749101016453316, 8.3282619715340908460311327395, 8.621027301676920779348929163006, 9.6231270143363776256573917546, 10.56097860787005094735433277626, 11.6242649918860212740541206657, 12.30419106409639494632426295482, 13.24230435724803667784541619943, 14.15614085664263102045088058133, 15.673757188440918847403331889462, 16.413545165844927996831248024275, 17.13137801393893508262710176256, 17.85095749724363192355950636620, 18.98900077885883420352562975053, 19.65276323300471396734353977082, 20.41873960065842011879871887814, 21.4191232467000730354950296038, 21.729420866183330653185802256531, 23.16936444602014520524546037265, 24.16097416605368664698333351124