L(s) = 1 | + (0.460 + 0.887i)2-s + (−0.575 + 0.817i)4-s + (0.237 − 0.971i)5-s + (−0.990 − 0.134i)8-s + (0.971 − 0.237i)10-s + (0.995 + 0.0970i)11-s + (0.738 − 0.674i)13-s + (−0.337 − 0.941i)16-s + (−0.344 − 0.938i)17-s + (−0.629 − 0.777i)19-s + (0.657 + 0.753i)20-s + (0.372 + 0.928i)22-s + (0.193 − 0.981i)23-s + (−0.887 − 0.460i)25-s + (0.938 + 0.344i)26-s + ⋯ |
L(s) = 1 | + (0.460 + 0.887i)2-s + (−0.575 + 0.817i)4-s + (0.237 − 0.971i)5-s + (−0.990 − 0.134i)8-s + (0.971 − 0.237i)10-s + (0.995 + 0.0970i)11-s + (0.738 − 0.674i)13-s + (−0.337 − 0.941i)16-s + (−0.344 − 0.938i)17-s + (−0.629 − 0.777i)19-s + (0.657 + 0.753i)20-s + (0.372 + 0.928i)22-s + (0.193 − 0.981i)23-s + (−0.887 − 0.460i)25-s + (0.938 + 0.344i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.274 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.274 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.332126812 - 1.005119917i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.332126812 - 1.005119917i\) |
\(L(1)\) |
\(\approx\) |
\(1.243750069 + 0.1497680675i\) |
\(L(1)\) |
\(\approx\) |
\(1.243750069 + 0.1497680675i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 + (0.460 + 0.887i)T \) |
| 5 | \( 1 + (0.237 - 0.971i)T \) |
| 11 | \( 1 + (0.995 + 0.0970i)T \) |
| 13 | \( 1 + (0.738 - 0.674i)T \) |
| 17 | \( 1 + (-0.344 - 0.938i)T \) |
| 19 | \( 1 + (-0.629 - 0.777i)T \) |
| 23 | \( 1 + (0.193 - 0.981i)T \) |
| 29 | \( 1 + (0.244 + 0.969i)T \) |
| 31 | \( 1 + (0.978 + 0.207i)T \) |
| 37 | \( 1 + (0.0149 - 0.999i)T \) |
| 43 | \( 1 + (-0.880 - 0.473i)T \) |
| 47 | \( 1 + (0.301 - 0.953i)T \) |
| 53 | \( 1 + (-0.985 + 0.171i)T \) |
| 59 | \( 1 + (-0.525 - 0.850i)T \) |
| 61 | \( 1 + (-0.323 + 0.946i)T \) |
| 67 | \( 1 + (-0.998 - 0.0523i)T \) |
| 71 | \( 1 + (0.969 + 0.244i)T \) |
| 73 | \( 1 + (0.149 + 0.988i)T \) |
| 79 | \( 1 + (0.258 - 0.965i)T \) |
| 83 | \( 1 + (-0.623 - 0.781i)T \) |
| 89 | \( 1 + (-0.976 - 0.215i)T \) |
| 97 | \( 1 + (-0.453 - 0.891i)T \) |
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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
−17.97199871493692857244157135282, −17.31897646932114879505240769479, −16.728107743519469568526872786181, −15.45225946668383337820301528364, −15.17069917534761886434571251853, −14.4012628521105175877209956425, −13.774909340429138875783193778047, −13.46326542326798045101824036956, −12.46167608357009727401581276348, −11.83327957287853752481754277724, −11.179477852106981132520370962, −10.81600558666684632789190185318, −9.866238995606442868829248655876, −9.55489342107806044024723933386, −8.6015058951656904281143538933, −7.949056250182363844822062836838, −6.662984461847154642692332371730, −6.29973569765185853603870386257, −5.81686813762775295850912441945, −4.588500252292838957310598013, −4.00220126000414601746211163080, −3.428500159002439473652112663386, −2.664809115286605387344658425566, −1.68506032923779906693237195445, −1.36788920701793964814340600085,
0.356331440795367735945662069496, 1.22185310693715517148645046968, 2.39486980118305878078612080402, 3.27462982448181769960055613806, 4.12414099305844637055640846469, 4.72726899108371521980966984174, 5.25792855579054546068861365391, 6.149465867153560315567367851307, 6.64871806020445095718714741725, 7.372366626736988057477128397365, 8.36619877277908080481579978281, 8.7648338942638590532243999832, 9.22066827250793591077750975488, 10.13379171960991524785241603861, 11.10212400153528584860551586278, 11.90810690100167367564332618479, 12.493632326879401140373771776633, 13.07478219912323458677038248673, 13.69085604226753015829345675245, 14.24555078280315970760846413026, 15.020638215192117508852967534883, 15.739048020661619925627408063473, 16.15952142788823358382738830536, 16.9210780559380399584743605608, 17.331783492370358991177377137