L(s) = 1 | + (−0.949 − 0.313i)2-s + (−0.927 + 0.373i)3-s + (0.803 + 0.595i)4-s + (−0.270 + 0.962i)5-s + (0.997 − 0.0637i)6-s + (0.326 + 0.945i)7-s + (−0.576 − 0.817i)8-s + (0.721 − 0.692i)9-s + (0.558 − 0.829i)10-s + (0.868 − 0.495i)11-s + (−0.967 − 0.252i)12-s + (−0.904 − 0.427i)13-s + (−0.0132 − 0.999i)14-s + (−0.108 − 0.994i)15-s + (0.290 + 0.956i)16-s + (−0.653 + 0.756i)17-s + ⋯ |
L(s) = 1 | + (−0.949 − 0.313i)2-s + (−0.927 + 0.373i)3-s + (0.803 + 0.595i)4-s + (−0.270 + 0.962i)5-s + (0.997 − 0.0637i)6-s + (0.326 + 0.945i)7-s + (−0.576 − 0.817i)8-s + (0.721 − 0.692i)9-s + (0.558 − 0.829i)10-s + (0.868 − 0.495i)11-s + (−0.967 − 0.252i)12-s + (−0.904 − 0.427i)13-s + (−0.0132 − 0.999i)14-s + (−0.108 − 0.994i)15-s + (0.290 + 0.956i)16-s + (−0.653 + 0.756i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.536 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.536 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1403433142 + 0.2554380570i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1403433142 + 0.2554380570i\) |
\(L(1)\) |
\(\approx\) |
\(0.4093045838 + 0.2024057278i\) |
\(L(1)\) |
\(\approx\) |
\(0.4093045838 + 0.2024057278i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 4729 | \( 1 \) |
good | 2 | \( 1 + (-0.949 - 0.313i)T \) |
| 3 | \( 1 + (-0.927 + 0.373i)T \) |
| 5 | \( 1 + (-0.270 + 0.962i)T \) |
| 7 | \( 1 + (0.326 + 0.945i)T \) |
| 11 | \( 1 + (0.868 - 0.495i)T \) |
| 13 | \( 1 + (-0.904 - 0.427i)T \) |
| 17 | \( 1 + (-0.653 + 0.756i)T \) |
| 19 | \( 1 + (0.280 + 0.959i)T \) |
| 23 | \( 1 + (-0.924 + 0.380i)T \) |
| 29 | \( 1 + (0.313 + 0.949i)T \) |
| 31 | \( 1 + (-0.999 - 0.0398i)T \) |
| 37 | \( 1 + (-0.545 + 0.838i)T \) |
| 41 | \( 1 + (0.994 + 0.103i)T \) |
| 43 | \( 1 + (0.0743 + 0.997i)T \) |
| 47 | \( 1 + (-0.997 - 0.0663i)T \) |
| 53 | \( 1 + (-0.272 + 0.962i)T \) |
| 59 | \( 1 + (-0.735 + 0.677i)T \) |
| 61 | \( 1 + (-0.641 - 0.767i)T \) |
| 67 | \( 1 + (0.979 - 0.200i)T \) |
| 71 | \( 1 + (-0.486 + 0.873i)T \) |
| 73 | \( 1 + (-0.531 + 0.846i)T \) |
| 79 | \( 1 + (0.866 + 0.5i)T \) |
| 83 | \( 1 + (0.833 - 0.551i)T \) |
| 89 | \( 1 + (0.0212 + 0.999i)T \) |
| 97 | \( 1 + (-0.326 - 0.945i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.73583398952471750736805976594, −17.11875630968221503629995610064, −16.524664508321232429776557195915, −16.09711199331499177586304323494, −15.353625553796504965952795789161, −14.37163506510457734093800280728, −13.74276342284125955821878552135, −12.83462996242196847596113317246, −12.01125897139309804208036165026, −11.64555461226183456824116023426, −10.98426054299007933850190843820, −10.17709320979822820082348933988, −9.4430038853349255019434056379, −8.9747229100567789304059650428, −7.84035560326458105887277339935, −7.4325892238638244131395582490, −6.82827682772062111841515167447, −6.15842839175188667928450925733, −5.04886155818884256611800086448, −4.72044452946347117046556097892, −3.878892591118947373017283051184, −2.19317597677716883556791478385, −1.71880561270592047689644186614, −0.69041349388206242382507074271, −0.18204988430393649384703694490,
1.32236855567404512502465938602, 2.00750897691968281587944618271, 3.06042261321168018377445982517, 3.64579688457892909777644188161, 4.55794781744411420253647527827, 5.75649268907535619928790550460, 6.15134815269051404272588870089, 6.87634549315609215496586042726, 7.680811614925141534633190131841, 8.35905696014191282753910897819, 9.2669932990122439193082322446, 9.831170047923058652900752214066, 10.529260194365945771255414797986, 11.15258474519049206774599583154, 11.61383967000023473186419155785, 12.286521684743352251382296183560, 12.69702377479759415930281314660, 14.20114716000565860099312955349, 14.823257167012663093282327388880, 15.4076282325611760077530858740, 16.0386961518144755447353900189, 16.723544656410913637689846051088, 17.40756221933844041238991962583, 18.054562189011696977380856158192, 18.33515254557094451581942604136