L(s) = 1 | + 1.73i·5-s − 1.73i·11-s − 3.46i·17-s − 2·19-s + 2.00·25-s − 9·29-s + 5·31-s + 10·37-s + 10.3i·41-s + 3.46i·43-s − 12·47-s + 9·53-s + 2.99·55-s + 9·59-s − 13.8i·67-s + ⋯ |
L(s) = 1 | + 0.774i·5-s − 0.522i·11-s − 0.840i·17-s − 0.458·19-s + 0.400·25-s − 1.67·29-s + 0.898·31-s + 1.64·37-s + 1.62i·41-s + 0.528i·43-s − 1.75·47-s + 1.23·53-s + 0.404·55-s + 1.17·59-s − 1.69i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.754850734\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.754850734\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 1.73iT - 5T^{2} \) |
| 11 | \( 1 + 1.73iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 - 10.3iT - 41T^{2} \) |
| 43 | \( 1 - 3.46iT - 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 - 9T + 53T^{2} \) |
| 59 | \( 1 - 9T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 13.8iT - 67T^{2} \) |
| 71 | \( 1 + 13.8iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 + 5.19iT - 79T^{2} \) |
| 83 | \( 1 + 3T + 83T^{2} \) |
| 89 | \( 1 + 3.46iT - 89T^{2} \) |
| 97 | \( 1 + 19.0iT - 97T^{2} \) |
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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
−7.86137978480158995060689498092, −7.21004650151500801986709774532, −6.43784893011807652739358521736, −6.00949109911626058410884031094, −5.01991687078121987870403959996, −4.35256486651412046606543450964, −3.30968187324325453840416597687, −2.84277555227717277885340193756, −1.83920404413216069526400203015, −0.55657533732187914545903509249,
0.816983947327139113656875989110, 1.82481290133913226245253154788, 2.63806026196075987504731752008, 3.91402477584844330865456522900, 4.23348982272405747351825015179, 5.26864435459295427135276181266, 5.69289053324231869194313502211, 6.67190983449897155422949657068, 7.23328218676107066828085211221, 8.167720925401891046948706943258