L(s) = 1 | + (0.986 − 0.162i)3-s + (−0.227 − 0.973i)5-s + (0.946 − 0.321i)9-s + (0.412 − 0.910i)11-s + (0.290 + 0.956i)13-s + (−0.382 − 0.923i)15-s + (0.608 + 0.793i)17-s + (−0.0327 − 0.999i)19-s + (−0.442 + 0.896i)23-s + (−0.896 + 0.442i)25-s + (0.881 − 0.471i)27-s + (−0.0980 + 0.995i)29-s + (0.258 + 0.965i)31-s + (0.258 − 0.965i)33-s + (−0.528 + 0.849i)37-s + ⋯ |
L(s) = 1 | + (0.986 − 0.162i)3-s + (−0.227 − 0.973i)5-s + (0.946 − 0.321i)9-s + (0.412 − 0.910i)11-s + (0.290 + 0.956i)13-s + (−0.382 − 0.923i)15-s + (0.608 + 0.793i)17-s + (−0.0327 − 0.999i)19-s + (−0.442 + 0.896i)23-s + (−0.896 + 0.442i)25-s + (0.881 − 0.471i)27-s + (−0.0980 + 0.995i)29-s + (0.258 + 0.965i)31-s + (0.258 − 0.965i)33-s + (−0.528 + 0.849i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.437367875 + 1.171231649i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.437367875 + 1.171231649i\) |
\(L(1)\) |
\(\approx\) |
\(1.456517117 - 0.09739751744i\) |
\(L(1)\) |
\(\approx\) |
\(1.456517117 - 0.09739751744i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.986 - 0.162i)T \) |
| 5 | \( 1 + (-0.227 - 0.973i)T \) |
| 11 | \( 1 + (0.412 - 0.910i)T \) |
| 13 | \( 1 + (0.290 + 0.956i)T \) |
| 17 | \( 1 + (0.608 + 0.793i)T \) |
| 19 | \( 1 + (-0.0327 - 0.999i)T \) |
| 23 | \( 1 + (-0.442 + 0.896i)T \) |
| 29 | \( 1 + (-0.0980 + 0.995i)T \) |
| 31 | \( 1 + (0.258 + 0.965i)T \) |
| 37 | \( 1 + (-0.528 + 0.849i)T \) |
| 41 | \( 1 + (-0.831 + 0.555i)T \) |
| 43 | \( 1 + (-0.634 + 0.773i)T \) |
| 47 | \( 1 + (0.130 + 0.991i)T \) |
| 53 | \( 1 + (0.910 + 0.412i)T \) |
| 59 | \( 1 + (0.683 + 0.729i)T \) |
| 61 | \( 1 + (-0.935 - 0.352i)T \) |
| 67 | \( 1 + (0.162 + 0.986i)T \) |
| 71 | \( 1 + (0.195 + 0.980i)T \) |
| 73 | \( 1 + (-0.659 + 0.751i)T \) |
| 79 | \( 1 + (0.793 + 0.608i)T \) |
| 83 | \( 1 + (0.471 - 0.881i)T \) |
| 89 | \( 1 + (-0.997 - 0.0654i)T \) |
| 97 | \( 1 + (-0.707 + 0.707i)T \) |
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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
−19.98717830645401209097251461150, −19.16957515207744947479413801993, −18.49430050188382770523168611728, −18.03002009478381310295140966115, −16.93368574317664043892093087893, −16.03584286946059842558522165708, −15.10104473148111125037309785662, −14.969111295911805268157887510105, −14.02312542271328317528806551137, −13.50249437060622581029828126994, −12.37806433720413258733310421921, −11.842904933137702720419098158423, −10.5659591676198573856984019354, −10.14535351315690277455779019819, −9.46936343950578204786909440651, −8.36311448972206423513949078466, −7.7435335845195623598859848196, −7.107409507737104291845218506684, −6.212234873328851100050881968615, −5.12275892587302359458376020930, −3.95760373201946202616032625202, −3.53263395024336832933918768599, −2.50259438220974349537042956781, −1.8792597985189036139131622966, −0.40200036858900167139224857502,
1.21184760372892930304360541374, 1.50174285010112100133021005692, 2.913837980231829002624101337408, 3.68136078683617251162777692176, 4.40209298353430151709535081349, 5.36039577813458851262323657645, 6.42818904479438034665607540374, 7.22559513034987522033246141383, 8.26708063915097164109069773557, 8.65451051471603560807779485620, 9.29893302424192328672638692600, 10.1329460222978916209219015329, 11.293150215349722660849385544214, 11.98590333953335083904412116520, 12.806366099264228578577378472908, 13.54539437317484422802050306877, 14.03005792835627752948865194154, 14.919366618495784671171960125932, 15.746109543578143071391453772202, 16.35589490189911100440742808776, 17.05803181476749050742019451867, 18.012314957095119836257480909590, 18.94728600997599744143666871392, 19.49047229097095226462311943666, 19.96592133575141759056272996518