L(s) = 1 | + 1.58i·3-s + 3.58i·5-s + 1.89i·7-s + 0.496·9-s − 1.56i·11-s + 4.45·13-s − 5.67·15-s + (4.09 − 0.481i)17-s − 5.20·19-s − 2.99·21-s − 3.19i·23-s − 7.84·25-s + 5.53i·27-s + 5.05i·29-s + 3.34i·31-s + ⋯ |
L(s) = 1 | + 0.913i·3-s + 1.60i·5-s + 0.715i·7-s + 0.165·9-s − 0.471i·11-s + 1.23·13-s − 1.46·15-s + (0.993 − 0.116i)17-s − 1.19·19-s − 0.653·21-s − 0.665i·23-s − 1.56·25-s + 1.06i·27-s + 0.938i·29-s + 0.600i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.760174722\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.760174722\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + (-4.09 + 0.481i)T \) |
| 59 | \( 1 - T \) |
good | 3 | \( 1 - 1.58iT - 3T^{2} \) |
| 5 | \( 1 - 3.58iT - 5T^{2} \) |
| 7 | \( 1 - 1.89iT - 7T^{2} \) |
| 11 | \( 1 + 1.56iT - 11T^{2} \) |
| 13 | \( 1 - 4.45T + 13T^{2} \) |
| 19 | \( 1 + 5.20T + 19T^{2} \) |
| 23 | \( 1 + 3.19iT - 23T^{2} \) |
| 29 | \( 1 - 5.05iT - 29T^{2} \) |
| 31 | \( 1 - 3.34iT - 31T^{2} \) |
| 37 | \( 1 - 10.4iT - 37T^{2} \) |
| 41 | \( 1 + 9.35iT - 41T^{2} \) |
| 43 | \( 1 + 7.48T + 43T^{2} \) |
| 47 | \( 1 + 8.79T + 47T^{2} \) |
| 53 | \( 1 + 2.62T + 53T^{2} \) |
| 61 | \( 1 - 12.4iT - 61T^{2} \) |
| 67 | \( 1 - 1.37T + 67T^{2} \) |
| 71 | \( 1 - 9.38iT - 71T^{2} \) |
| 73 | \( 1 + 0.343iT - 73T^{2} \) |
| 79 | \( 1 - 0.0938iT - 79T^{2} \) |
| 83 | \( 1 + 5.61T + 83T^{2} \) |
| 89 | \( 1 - 7.62T + 89T^{2} \) |
| 97 | \( 1 + 17.3iT - 97T^{2} \) |
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
−8.665061298049885973221523540821, −8.391959908913373493013263578400, −7.16760820149429083088022725783, −6.59670001845697499212677438975, −5.93459370300242289941961754214, −5.13331659050782208334291663061, −4.07818842729413885789183366666, −3.35674273016645992748149233899, −2.85139053767975192256533064312, −1.59351717802564258831615452407,
0.52335519025261113708572628172, 1.35811224471862550452790234394, 1.97287638304118794644023227122, 3.63269050279311661805699113389, 4.25575247468047183650876227230, 5.01793120637932784377292347105, 5.98727050532105110165393863512, 6.52815332423783660642798860176, 7.57801873720087839138793526658, 7.999150434416575215080452514802