L(s) = 1 | − 2i·3-s − i·5-s − 2i·7-s − 9-s + 2·13-s − 2·15-s + 4·19-s − 4·21-s − 6i·23-s − 25-s − 4i·27-s + 6i·29-s − 4i·31-s − 2·35-s + 2i·37-s + ⋯ |
L(s) = 1 | − 1.15i·3-s − 0.447i·5-s − 0.755i·7-s − 0.333·9-s + 0.554·13-s − 0.516·15-s + 0.917·19-s − 0.872·21-s − 1.25i·23-s − 0.200·25-s − 0.769i·27-s + 1.11i·29-s − 0.718i·31-s − 0.338·35-s + 0.328i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.937505363\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.937505363\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 17 | \( 1 \) |
good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 6iT - 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 8iT - 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−7.54550727696115930804371067082, −7.27855198407082602559688554469, −6.42964571963659536158206486889, −5.85326470100329636488713623089, −4.85827161112709190911421409391, −4.13598502224233865669736387375, −3.21686565216667850936633096004, −2.15625909029528111671861946318, −1.24702266708740534038997214756, −0.55378584897632678778180131710,
1.32115725402431437071387720035, 2.54698066709630898551387348849, 3.32920445921698485353125905042, 3.97450644175189896761544431501, 4.79908745620068011328590817665, 5.59115023153883955987047102669, 6.03874362819015385246231840984, 7.09424566657196818893143442697, 7.73276061797015930620379998943, 8.616773272986044176881022167148