L(s) = 1 | + (−0.939 + 0.342i)3-s + (0.984 + 0.173i)7-s + (0.766 − 0.642i)9-s + (0.866 − 0.5i)11-s + (0.642 − 0.766i)13-s + (0.766 − 0.642i)17-s + (−0.939 + 0.342i)19-s + (−0.984 + 0.173i)21-s + (−0.5 + 0.866i)23-s + (−0.5 + 0.866i)27-s + (0.5 + 0.866i)29-s − i·31-s + (−0.642 + 0.766i)33-s + (−0.342 + 0.939i)39-s + (0.766 + 0.642i)41-s + ⋯ |
L(s) = 1 | + (−0.939 + 0.342i)3-s + (0.984 + 0.173i)7-s + (0.766 − 0.642i)9-s + (0.866 − 0.5i)11-s + (0.642 − 0.766i)13-s + (0.766 − 0.642i)17-s + (−0.939 + 0.342i)19-s + (−0.984 + 0.173i)21-s + (−0.5 + 0.866i)23-s + (−0.5 + 0.866i)27-s + (0.5 + 0.866i)29-s − i·31-s + (−0.642 + 0.766i)33-s + (−0.342 + 0.939i)39-s + (0.766 + 0.642i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.583768689 + 0.3167053835i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.583768689 + 0.3167053835i\) |
\(L(1)\) |
\(\approx\) |
\(1.026691981 + 0.09122490450i\) |
\(L(1)\) |
\(\approx\) |
\(1.026691981 + 0.09122490450i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 3 | \( 1 + (-0.939 + 0.342i)T \) |
| 7 | \( 1 + (0.984 + 0.173i)T \) |
| 11 | \( 1 + (0.866 - 0.5i)T \) |
| 13 | \( 1 + (0.642 - 0.766i)T \) |
| 17 | \( 1 + (0.766 - 0.642i)T \) |
| 19 | \( 1 + (-0.939 + 0.342i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 - iT \) |
| 41 | \( 1 + (0.766 + 0.642i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (0.866 + 0.5i)T \) |
| 53 | \( 1 + (0.173 + 0.984i)T \) |
| 59 | \( 1 + (-0.173 - 0.984i)T \) |
| 61 | \( 1 + (0.766 + 0.642i)T \) |
| 67 | \( 1 + (-0.173 + 0.984i)T \) |
| 71 | \( 1 + (0.939 - 0.342i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (-0.984 - 0.173i)T \) |
| 83 | \( 1 + (-0.766 + 0.642i)T \) |
| 89 | \( 1 + (0.984 - 0.173i)T \) |
| 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
−18.93940249359609829181463532649, −18.28750876643130146607402815545, −17.47756254612490380835255800177, −17.14439042074865800947333671046, −16.46150830711940325100961796364, −15.63784516913069039026761163533, −14.75867541765017659022013892797, −14.13856539247846031671271009396, −13.441244589075675867443655829108, −12.32981540388258097946854992945, −12.11385106075539180856005093684, −11.25039423374221384334675182615, −10.64917521999346174701351848211, −10.02009788381938483523831676348, −8.85896419932823452909480134401, −8.2701571716396555341048885976, −7.3370321590940357020827549170, −6.64597706370629274130612527935, −6.045523946821797961421242333, −5.14703505972749525230753718152, −4.300967929756532715569148872505, −3.911056931848002410231753163188, −2.20890196411609874634008641214, −1.656087225867165489076210849842, −0.763820564534871473797474345187,
0.92779518224398735620891049009, 1.460044768619122274734058546431, 2.80610080950453896509198996923, 3.86745238366115062343878853679, 4.393465493520117131748291339957, 5.462059293525946883572597048969, 5.80223353124315226225898144805, 6.647916022960036657946279798101, 7.640270288066224581956231720007, 8.30603201241628229927208936717, 9.20928086045996020669195560566, 9.9384468104978932456743186582, 10.8503308010114194279365332846, 11.242346232773207484070743713868, 11.94381713547925753695972323216, 12.57727352455042908697961459939, 13.50093502634264370341750353661, 14.403027052760141287337915861674, 14.89473530095690273350874069459, 15.81361134871303093794509257521, 16.323231250074748671572928836875, 17.21333222696008650118761185042, 17.55322629210513172469942056012, 18.39242334739435535322170435878, 18.8664651425229276936424070986