L(s) = 1 | + (0.654 + 0.755i)3-s + (−0.142 − 0.989i)5-s + (−0.415 + 0.909i)7-s + (−0.142 + 0.989i)9-s + (0.959 + 0.281i)11-s + (0.415 + 0.909i)13-s + (0.654 − 0.755i)15-s + (0.841 + 0.540i)17-s + (−0.841 + 0.540i)19-s + (−0.959 + 0.281i)21-s + (−0.959 + 0.281i)25-s + (−0.841 + 0.540i)27-s + (0.841 + 0.540i)29-s + (0.654 − 0.755i)31-s + (0.415 + 0.909i)33-s + ⋯ |
L(s) = 1 | + (0.654 + 0.755i)3-s + (−0.142 − 0.989i)5-s + (−0.415 + 0.909i)7-s + (−0.142 + 0.989i)9-s + (0.959 + 0.281i)11-s + (0.415 + 0.909i)13-s + (0.654 − 0.755i)15-s + (0.841 + 0.540i)17-s + (−0.841 + 0.540i)19-s + (−0.959 + 0.281i)21-s + (−0.959 + 0.281i)25-s + (−0.841 + 0.540i)27-s + (0.841 + 0.540i)29-s + (0.654 − 0.755i)31-s + (0.415 + 0.909i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.178 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.178 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.482850506 + 1.237901014i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.482850506 + 1.237901014i\) |
\(L(1)\) |
\(\approx\) |
\(1.234806650 + 0.4499696598i\) |
\(L(1)\) |
\(\approx\) |
\(1.234806650 + 0.4499696598i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (0.654 + 0.755i)T \) |
| 5 | \( 1 + (-0.142 - 0.989i)T \) |
| 7 | \( 1 + (-0.415 + 0.909i)T \) |
| 11 | \( 1 + (0.959 + 0.281i)T \) |
| 13 | \( 1 + (0.415 + 0.909i)T \) |
| 17 | \( 1 + (0.841 + 0.540i)T \) |
| 19 | \( 1 + (-0.841 + 0.540i)T \) |
| 29 | \( 1 + (0.841 + 0.540i)T \) |
| 31 | \( 1 + (0.654 - 0.755i)T \) |
| 37 | \( 1 + (-0.142 + 0.989i)T \) |
| 41 | \( 1 + (-0.142 - 0.989i)T \) |
| 43 | \( 1 + (0.654 + 0.755i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (0.415 - 0.909i)T \) |
| 59 | \( 1 + (-0.415 - 0.909i)T \) |
| 61 | \( 1 + (-0.654 + 0.755i)T \) |
| 67 | \( 1 + (0.959 - 0.281i)T \) |
| 71 | \( 1 + (0.959 - 0.281i)T \) |
| 73 | \( 1 + (0.841 - 0.540i)T \) |
| 79 | \( 1 + (-0.415 - 0.909i)T \) |
| 83 | \( 1 + (0.142 - 0.989i)T \) |
| 89 | \( 1 + (-0.654 - 0.755i)T \) |
| 97 | \( 1 + (-0.142 - 0.989i)T \) |
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
−30.15933219976261892424418740801, −29.28917705495921396471488379712, −27.53528142407963806451377639232, −26.569804503705975491826293658526, −25.66307792313923449113284150422, −24.82085710264608321718857401773, −23.33541636677874739860753306477, −22.83427642905755531234694153647, −21.290968109403925262026622358166, −19.89479795551011078888310339549, −19.3357717421579243683544038596, −18.191789182250434912214358466146, −17.145642601123555369697815278124, −15.54008190068117279917674771839, −14.33999362939158907279194055617, −13.66622798632950824856083043972, −12.35810584695332609437034077287, −10.98649715865278574176919835980, −9.76026694773889611112726002428, −8.23627068585385023637207486259, −7.10226991540507293088382562564, −6.2674756651481775361984182590, −3.81891792577177107555900738972, −2.82210653992362497672918025882, −0.87717114495869139512230307560,
1.805158219570453969461263840657, 3.60894831769389012065628684042, 4.72315324013003901460992092215, 6.19834667240852470333108899338, 8.23517664762689935531910975023, 9.00871698108334970166974634900, 9.93785120334933841140141319823, 11.68316401139650586337854857042, 12.71523947184592322121241527124, 14.11785242625286632214825665589, 15.18949644898338970558069097574, 16.243866545451809950299219681506, 17.028458924110294085872123181419, 18.94401115026260663967363110231, 19.66177319849228976806849141092, 20.9304729573463058212422133580, 21.5512824023151398678422977169, 22.80648741837304741462597929198, 24.23955705088420212328005564985, 25.28179651644809580962494275312, 25.91921595603587492949505833774, 27.54415688819423674930707557013, 27.917683693840366426421659327161, 29.0568040402808373600245435355, 30.66504213941491342840255021259