L(s) = 1 | + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)5-s + (−0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (0.309 − 0.951i)10-s + (−0.5 + 1.53i)13-s + (0.309 − 0.951i)16-s + (−0.5 − 0.363i)17-s + 0.999·18-s + 0.999·20-s + (0.309 + 0.951i)25-s − 1.61·26-s + (−0.5 + 0.363i)29-s + 32-s + ⋯ |
L(s) = 1 | + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)5-s + (−0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (0.309 − 0.951i)10-s + (−0.5 + 1.53i)13-s + (0.309 − 0.951i)16-s + (−0.5 − 0.363i)17-s + 0.999·18-s + 0.999·20-s + (0.309 + 0.951i)25-s − 1.61·26-s + (−0.5 + 0.363i)29-s + 32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5534260427\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5534260427\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.309 - 0.951i)T \) |
| 5 | \( 1 + (0.809 + 0.587i)T \) |
good | 3 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.53082763279916007951704825813, −13.33715435256026911803134895286, −12.30978471057418900351284262357, −11.56782324166309553129123243505, −9.499591714021149766685915326899, −8.795965928933043041405356203034, −7.44260981790956792459665870802, −6.52341597470470667371308755766, −4.84001626020136688398360006484, −3.82709231667603315271129119450,
2.66152548206980973020188319867, 4.14002826679925869875833378269, 5.52389601233870343120666487150, 7.40735181615435052329967335639, 8.524044957078315048199567134733, 10.21714847714142870445182399272, 10.72134589933074432226622561992, 11.84084510380340478198286773064, 12.83333147423805482282199149653, 13.73966495852089220025482417049