L(s) = 1 | + (0.723 + 0.690i)2-s + (0.0475 + 0.998i)4-s + (−0.981 + 0.189i)5-s + (−0.580 + 0.814i)7-s + (−0.654 + 0.755i)8-s + (−0.841 − 0.540i)10-s + (−0.0475 − 0.998i)13-s + (−0.981 + 0.189i)14-s + (−0.995 + 0.0950i)16-s + (−0.142 + 0.989i)17-s + (0.142 + 0.989i)19-s + (−0.235 − 0.971i)20-s + (−0.580 − 0.814i)23-s + (0.928 − 0.371i)25-s + (0.654 − 0.755i)26-s + ⋯ |
L(s) = 1 | + (0.723 + 0.690i)2-s + (0.0475 + 0.998i)4-s + (−0.981 + 0.189i)5-s + (−0.580 + 0.814i)7-s + (−0.654 + 0.755i)8-s + (−0.841 − 0.540i)10-s + (−0.0475 − 0.998i)13-s + (−0.981 + 0.189i)14-s + (−0.995 + 0.0950i)16-s + (−0.142 + 0.989i)17-s + (0.142 + 0.989i)19-s + (−0.235 − 0.971i)20-s + (−0.580 − 0.814i)23-s + (0.928 − 0.371i)25-s + (0.654 − 0.755i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.122 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.122 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1758722800 + 0.1988741385i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1758722800 + 0.1988741385i\) |
\(L(1)\) |
\(\approx\) |
\(0.7357283863 + 0.5503662447i\) |
\(L(1)\) |
\(\approx\) |
\(0.7357283863 + 0.5503662447i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.723 + 0.690i)T \) |
| 5 | \( 1 + (-0.981 + 0.189i)T \) |
| 7 | \( 1 + (-0.580 + 0.814i)T \) |
| 13 | \( 1 + (-0.0475 - 0.998i)T \) |
| 17 | \( 1 + (-0.142 + 0.989i)T \) |
| 19 | \( 1 + (0.142 + 0.989i)T \) |
| 23 | \( 1 + (-0.580 - 0.814i)T \) |
| 29 | \( 1 + (-0.786 + 0.618i)T \) |
| 31 | \( 1 + (-0.888 + 0.458i)T \) |
| 37 | \( 1 + (0.841 + 0.540i)T \) |
| 41 | \( 1 + (0.235 - 0.971i)T \) |
| 43 | \( 1 + (-0.981 - 0.189i)T \) |
| 47 | \( 1 + (-0.235 - 0.971i)T \) |
| 53 | \( 1 + (-0.415 - 0.909i)T \) |
| 59 | \( 1 + (-0.723 + 0.690i)T \) |
| 61 | \( 1 + (-0.235 - 0.971i)T \) |
| 67 | \( 1 + (0.235 - 0.971i)T \) |
| 71 | \( 1 + (0.142 + 0.989i)T \) |
| 73 | \( 1 + (-0.415 + 0.909i)T \) |
| 79 | \( 1 + (0.327 - 0.945i)T \) |
| 83 | \( 1 + (0.580 - 0.814i)T \) |
| 89 | \( 1 + (0.142 - 0.989i)T \) |
| 97 | \( 1 + (0.981 + 0.189i)T \) |
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
−20.72990533535825311805870932860, −20.045624606485987282537337933840, −19.600587331486899506524586926782, −18.86612025467895486426483534722, −18.01272166963200336200230132796, −16.69184501430118567353467575469, −16.10502647943446826849769454387, −15.3447690642532740835894531224, −14.46590836902583847415680369418, −13.58471146950943250414439862755, −13.0794531855633483058893428230, −12.077637629561973148141482026675, −11.361835569344885455470690903760, −10.92846397348374048498130422580, −9.545923749208864746211994052898, −9.30048920561409383188918641714, −7.71501873585959444544376015623, −7.04129832991820617788342011689, −6.13777811112291330268911547836, −4.88712159549541754435952512274, −4.23697577492433207685673586348, −3.52889573832901280672654308910, −2.58738935012910256810723227900, −1.27176929169272071846090844491, −0.08683367695556053067852886112,
2.10119669923722142477570042073, 3.3186451499946911991247582647, 3.67627293836824878182545733424, 4.87720216490247533868024410309, 5.76387483721851693663706069490, 6.464804870684530030650345284917, 7.45475563483261640507626799727, 8.222129175681129806686578612580, 8.79369070139897638128456586819, 10.13212955218727194582924813426, 11.09101377327418926597732608088, 12.11190399443829482450707657933, 12.5424855739986760807610455913, 13.212686359646901648289915151768, 14.58947024124036322824101345602, 14.88205320772991720668795426041, 15.71948963830489480311688837694, 16.28418159356396734270798410441, 17.06434077944782095362882312444, 18.254035833049768572820372749294, 18.69014651647889044344475371536, 19.90738461820879814390895872206, 20.36238886057863313979069890155, 21.565564007718910384197662309257, 22.186001123208075802617693190063