| L(s) = 1 | + 6.40i·3-s + 38.0·5-s − 18.5i·7-s + 39.9·9-s − 70.7i·11-s − 59.7·13-s + 243. i·15-s + 359.·17-s − 24.0i·19-s + 118.·21-s + 985. i·23-s + 820.·25-s + 774. i·27-s + 1.11e3·29-s − 656. i·31-s + ⋯ |
| L(s) = 1 | + 0.711i·3-s + 1.52·5-s − 0.377i·7-s + 0.493·9-s − 0.584i·11-s − 0.353·13-s + 1.08i·15-s + 1.24·17-s − 0.0664i·19-s + 0.269·21-s + 1.86i·23-s + 1.31·25-s + 1.06i·27-s + 1.32·29-s − 0.683i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.29328 + 0.614484i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.29328 + 0.614484i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 18.5iT \) |
| good | 3 | \( 1 - 6.40iT - 81T^{2} \) |
| 5 | \( 1 - 38.0T + 625T^{2} \) |
| 11 | \( 1 + 70.7iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 59.7T + 2.85e4T^{2} \) |
| 17 | \( 1 - 359.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 24.0iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 985. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.11e3T + 7.07e5T^{2} \) |
| 31 | \( 1 + 656. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.92e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 304.T + 2.82e6T^{2} \) |
| 43 | \( 1 - 1.21e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 3.31e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 57.6T + 7.89e6T^{2} \) |
| 59 | \( 1 - 4.47e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 4.63e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 2.60e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 4.94e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 7.89e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 9.56e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 5.92e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 4.47e3T + 6.27e7T^{2} \) |
| 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
−13.31469215394056087948563044607, −11.93693768669022769293058841059, −10.43725966166206873717761315250, −9.970760185952206732981365063439, −9.056290530821660942547673498943, −7.42102094686300815228492278709, −5.98545067072936084656715873980, −4.99455657785465050773522267191, −3.34301926756322860235705931173, −1.48139320946916739215590598300,
1.37072429663287437397662562212, 2.55122830176367713256311980552, 4.88734853034560212540703613161, 6.14480596155045786510857467568, 7.06595157629523807428336260952, 8.499340915191872362439819108809, 9.806994880735296949132243295703, 10.38783963896545365328566604630, 12.32596075748781600560189193959, 12.61617395700959228555626989244
