L(s) = 1 | + (0.973 − 0.230i)2-s + (0.893 − 0.448i)4-s + (0.597 + 0.802i)5-s + (−0.835 + 0.549i)7-s + (0.766 − 0.642i)8-s + (0.766 + 0.642i)10-s + (−0.993 − 0.116i)11-s + (−0.286 − 0.957i)13-s + (−0.686 + 0.727i)14-s + (0.597 − 0.802i)16-s + (0.173 − 0.984i)17-s + (0.173 + 0.984i)19-s + (0.893 + 0.448i)20-s + (−0.993 + 0.116i)22-s + (−0.835 − 0.549i)23-s + ⋯ |
L(s) = 1 | + (0.973 − 0.230i)2-s + (0.893 − 0.448i)4-s + (0.597 + 0.802i)5-s + (−0.835 + 0.549i)7-s + (0.766 − 0.642i)8-s + (0.766 + 0.642i)10-s + (−0.993 − 0.116i)11-s + (−0.286 − 0.957i)13-s + (−0.686 + 0.727i)14-s + (0.597 − 0.802i)16-s + (0.173 − 0.984i)17-s + (0.173 + 0.984i)19-s + (0.893 + 0.448i)20-s + (−0.993 + 0.116i)22-s + (−0.835 − 0.549i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.623567761 - 0.06300182309i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.623567761 - 0.06300182309i\) |
\(L(1)\) |
\(\approx\) |
\(1.650399504 - 0.07156843997i\) |
\(L(1)\) |
\(\approx\) |
\(1.650399504 - 0.07156843997i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + (0.973 - 0.230i)T \) |
| 5 | \( 1 + (0.597 + 0.802i)T \) |
| 7 | \( 1 + (-0.835 + 0.549i)T \) |
| 11 | \( 1 + (-0.993 - 0.116i)T \) |
| 13 | \( 1 + (-0.286 - 0.957i)T \) |
| 17 | \( 1 + (0.173 - 0.984i)T \) |
| 19 | \( 1 + (0.173 + 0.984i)T \) |
| 23 | \( 1 + (-0.835 - 0.549i)T \) |
| 29 | \( 1 + (-0.686 - 0.727i)T \) |
| 31 | \( 1 + (-0.0581 + 0.998i)T \) |
| 37 | \( 1 + (-0.939 + 0.342i)T \) |
| 41 | \( 1 + (0.973 + 0.230i)T \) |
| 43 | \( 1 + (0.396 + 0.918i)T \) |
| 47 | \( 1 + (-0.0581 - 0.998i)T \) |
| 53 | \( 1 + (-0.5 + 0.866i)T \) |
| 59 | \( 1 + (-0.993 + 0.116i)T \) |
| 61 | \( 1 + (0.893 + 0.448i)T \) |
| 67 | \( 1 + (-0.686 + 0.727i)T \) |
| 71 | \( 1 + (0.766 + 0.642i)T \) |
| 73 | \( 1 + (0.766 - 0.642i)T \) |
| 79 | \( 1 + (0.973 - 0.230i)T \) |
| 83 | \( 1 + (0.973 - 0.230i)T \) |
| 89 | \( 1 + (0.766 - 0.642i)T \) |
| 97 | \( 1 + (0.597 - 0.802i)T \) |
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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
−31.30865616916645049588247738282, −29.926600502791933541361149220967, −29.0702584220541991389659341715, −28.27059379779268454287572715109, −26.07199296724238850578858047861, −25.85517968961375494435031013468, −24.14952790892503858958692781257, −23.78860670003859176553086421770, −22.34226381545322554277350952331, −21.39997954846159124661520578707, −20.44147234424034250627594532035, −19.37098276622177645600415963280, −17.45892686970901663603041570977, −16.49754115880759754421758996854, −15.61391760755120163604967593870, −14.04660600323492631798815279232, −13.17617068766516122540723753170, −12.37964110232101448973285103294, −10.761469707202499829360733178136, −9.390277993418775098253405016350, −7.687448040669914946730215133599, −6.35120844369665879949068253931, −5.159131292285375575767094879007, −3.85713118016557444500966954739, −2.12652156444022943100733545988,
2.40911414521744057912588289218, 3.2943499622328760015142376625, 5.32545282084031423616502472636, 6.19450505012574488402066678964, 7.58209366769652586999848957206, 9.80914515239880851546838231624, 10.60030673078252877937333771063, 12.11581489157847531618208051144, 13.139073466883395744639931467813, 14.18682295174626141009446036693, 15.33428513180148347774956452463, 16.271207350810522720602246147895, 18.11488496943427504281887082373, 19.03448940590677206871190020674, 20.41601228867488936067057927224, 21.438792009116496059367427289909, 22.5202905178582941041306129857, 22.991359635490563353887072326455, 24.64743582634621786772682906209, 25.40669191423393596509449772985, 26.487783442083226522231887751625, 28.181415651877560799685829836343, 29.30337781833074885468723731493, 29.77157615615157276523676460043, 31.13215627610484408298584692250