L(s) = 1 | − 2.61i·2-s − 2.61·3-s − 4.85·4-s − 2.61i·5-s + 6.85i·6-s + i·7-s + 7.47i·8-s + 3.85·9-s − 6.85·10-s + 1.85i·11-s + 12.7·12-s + 2.61·14-s + 6.85i·15-s + 9.85·16-s + 1.47·17-s − 10.0i·18-s + ⋯ |
L(s) = 1 | − 1.85i·2-s − 1.51·3-s − 2.42·4-s − 1.17i·5-s + 2.79i·6-s + 0.377i·7-s + 2.64i·8-s + 1.28·9-s − 2.16·10-s + 0.559i·11-s + 3.66·12-s + 0.699·14-s + 1.76i·15-s + 2.46·16-s + 0.357·17-s − 2.37i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6851454920\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6851454920\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - iT \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 2.61iT - 2T^{2} \) |
| 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 + 2.61iT - 5T^{2} \) |
| 11 | \( 1 - 1.85iT - 11T^{2} \) |
| 17 | \( 1 - 1.47T + 17T^{2} \) |
| 19 | \( 1 - 1.85iT - 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 - 7.09T + 29T^{2} \) |
| 31 | \( 1 - 4.70iT - 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + 0.763iT - 41T^{2} \) |
| 43 | \( 1 + 12.5T + 43T^{2} \) |
| 47 | \( 1 + 2.23iT - 47T^{2} \) |
| 53 | \( 1 - 3.76T + 53T^{2} \) |
| 59 | \( 1 + 2.23iT - 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 12.7iT - 67T^{2} \) |
| 71 | \( 1 - 14.1iT - 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 6.70iT - 83T^{2} \) |
| 89 | \( 1 - 4.90iT - 89T^{2} \) |
| 97 | \( 1 + 18.8iT - 97T^{2} \) |
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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
−10.01334754546344607612683172777, −8.918754150586490194660209219330, −8.364188882902569677190534889660, −6.81742339835504787228716958904, −5.55804169668432054294894938583, −4.93235499036173288440157957498, −4.45123508975925128022489216101, −3.11190366982291820295760026603, −1.62644502057518434267803069052, −0.812109492845607523205066322376,
0.59295723061647359898440590104, 3.30271702574535043786347264086, 4.57600165197041968713615677515, 5.24670790399185567123742976358, 6.18511252449317536689309617781, 6.55284368690514802193420827397, 7.20655914977400169525899486687, 7.995967799851407947159370730143, 9.065701011558374629644367657984, 10.08556716707651377907038371688