L(s) = 1 | − i·3-s − 4i·7-s − 9-s − 4i·11-s − 2·13-s + (1 + 4i)17-s + 4·19-s − 4·21-s − 4i·23-s + 5·25-s + i·27-s − 4i·31-s − 4·33-s − 8i·37-s + 2i·39-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 1.51i·7-s − 0.333·9-s − 1.20i·11-s − 0.554·13-s + (0.242 + 0.970i)17-s + 0.917·19-s − 0.872·21-s − 0.834i·23-s + 25-s + 0.192i·27-s − 0.718i·31-s − 0.696·33-s − 1.31i·37-s + 0.320i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.392458825\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.392458825\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 17 | \( 1 + (-1 - 4i)T \) |
good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 8iT - 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 8iT - 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 12iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 16iT - 97T^{2} \) |
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
−8.219755801519699507705460993224, −7.44486484450179877122502191951, −6.99118947926346973047463441736, −6.10371856844874379457799870871, −5.37671718392059360697009925344, −4.27148727468441219632877604725, −3.58027834706863520847208800494, −2.62932445016937234369492917878, −1.27746190514501989079837182678, −0.45043395389257427566035179776,
1.57416822352247276781857461250, 2.72111079783153826558913884800, 3.22330237745520587823886317494, 4.79708760772068045718552473289, 4.91925910683796553955796754613, 5.80820234548126624288347657749, 6.77276000969561798510288412362, 7.51223519281866635742827379274, 8.366685458782787882605381157611, 9.140654236310545961783495048636