L(s) = 1 | + 2.82i·2-s − 8.00·4-s + 9.19i·5-s + 18.5·7-s − 22.6i·8-s − 25.9·10-s + 68.1i·11-s + 212.·13-s + 52.3i·14-s + 64.0·16-s − 340. i·17-s − 176.·19-s − 73.5i·20-s − 192.·22-s − 241. i·23-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.500·4-s + 0.367i·5-s + 0.377·7-s − 0.353i·8-s − 0.259·10-s + 0.563i·11-s + 1.25·13-s + 0.267i·14-s + 0.250·16-s − 1.17i·17-s − 0.488·19-s − 0.183i·20-s − 0.398·22-s − 0.456i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.043548408\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.043548408\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2.82iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 18.5T \) |
good | 5 | \( 1 - 9.19iT - 625T^{2} \) |
| 11 | \( 1 - 68.1iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 212.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 340. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 176.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 241. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 549. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 980.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 743.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 1.96e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.35e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 1.32e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 4.06e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 3.52e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 5.20e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 2.24e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 7.43e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 1.05e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 3.14e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 467. iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 2.86e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.58e4T + 8.85e7T^{2} \) |
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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
−10.91706765278967063778854725868, −9.997816347758387804356073616027, −8.918431726683895354778378058109, −8.166441024132307322244057959708, −7.07233914911651824723803718544, −6.37200906478356498865824436107, −5.16134815013939734633507293265, −4.19690710491412862262798159872, −2.78203060534989175820819682109, −1.05628378193242443144103720114,
0.74713715857206539878913860499, 1.86768549578199734363258936981, 3.38532514986569486799784649051, 4.32545283920799552824860322883, 5.54647836056254814071753726388, 6.53102306295946533827240259043, 8.232966662389218507201474020770, 8.508046866679911456528868949458, 9.688731448687857982890043786821, 10.74348452933952964352425451807