L(s) = 1 | − 3.95i·5-s − 4.77·7-s − 5.95i·11-s − 5.63i·17-s − 4.35·19-s + 3.33i·23-s − 10.6·25-s + 18.8i·35-s + 10.8·43-s + 0.845i·47-s + 15.8·49-s − 23.5·55-s + 15.1·61-s − 16.8·73-s + 28.4i·77-s + ⋯ |
L(s) = 1 | − 1.76i·5-s − 1.80·7-s − 1.79i·11-s − 1.36i·17-s − 1.00·19-s + 0.695i·23-s − 2.12·25-s + 3.19i·35-s + 1.65·43-s + 0.123i·47-s + 2.26·49-s − 3.17·55-s + 1.94·61-s − 1.96·73-s + 3.24i·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5666773354\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5666773354\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 4.35T \) |
good | 5 | \( 1 + 3.95iT - 5T^{2} \) |
| 7 | \( 1 + 4.77T + 7T^{2} \) |
| 11 | \( 1 + 5.95iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 5.63iT - 17T^{2} \) |
| 23 | \( 1 - 3.33iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 0.845iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 15.1T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 16.8T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 5.14iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.611025506801124206012052837427, −7.69460769003999616344568572296, −6.67217952949657642323863592554, −5.85672865973356957189600407017, −5.41559972546552638451734856441, −4.29459447822260081171838121278, −3.51935319289925344322605803965, −2.60928584283827658675864979218, −0.937823593254205181878926434019, −0.21922955850372402693366741329,
2.12223927202963058311630871797, 2.71806484957337314713512538682, 3.70966363281998011142306062997, 4.28239351266222366617840654647, 5.88049426447121337754709351152, 6.40024196163116676268577008519, 6.96519535629429471758075008788, 7.44680919032196651596862446466, 8.611326643463051349319724723000, 9.594343286569441468524558964257