| L(s) = 1 | + (−0.766 + 0.642i)2-s + (−1.15 + 1.29i)3-s + (0.173 − 0.984i)4-s + (−2.06 + 0.753i)5-s + (0.0560 − 1.73i)6-s + (0.173 + 0.984i)7-s + (0.500 + 0.866i)8-s + (−0.328 − 2.98i)9-s + (1.10 − 1.90i)10-s + (−5.37 − 1.95i)11-s + (1.06 + 1.36i)12-s + (2.61 + 2.19i)13-s + (−0.766 − 0.642i)14-s + (1.42 − 3.54i)15-s + (−0.939 − 0.342i)16-s + (1.91 − 3.31i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.454i)2-s + (−0.667 + 0.744i)3-s + (0.0868 − 0.492i)4-s + (−0.925 + 0.336i)5-s + (0.0228 − 0.706i)6-s + (0.0656 + 0.372i)7-s + (0.176 + 0.306i)8-s + (−0.109 − 0.993i)9-s + (0.348 − 0.603i)10-s + (−1.62 − 0.590i)11-s + (0.308 + 0.393i)12-s + (0.724 + 0.608i)13-s + (−0.204 − 0.171i)14-s + (0.366 − 0.914i)15-s + (−0.234 − 0.0855i)16-s + (0.463 − 0.803i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.169253 - 0.123491i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.169253 - 0.123491i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 3 | \( 1 + (1.15 - 1.29i)T \) |
| 7 | \( 1 + (-0.173 - 0.984i)T \) |
| good | 5 | \( 1 + (2.06 - 0.753i)T + (3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (5.37 + 1.95i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.61 - 2.19i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.91 + 3.31i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.761 + 1.31i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.898 + 5.09i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (0.792 - 0.665i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.63 + 9.29i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.40 + 2.43i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.73 - 1.45i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (6.53 + 2.37i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.242 - 1.37i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 7.06T + 53T^{2} \) |
| 59 | \( 1 + (4.62 - 1.68i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.390 + 2.21i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (9.77 + 8.20i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (2.19 - 3.79i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.0771 - 0.133i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8.88 - 7.45i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (2.52 - 2.12i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (6.18 + 10.7i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.77 + 2.83i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.15207139967587273968695189728, −10.35723649400829124170624868439, −9.345504905943491803469739467209, −8.368948644724548821960083610896, −7.53313782380466207764749548965, −6.34464367255399787015157315186, −5.42311870849140368373029995880, −4.36010973292901395980504918556, −2.94814229120724264391148254419, −0.18829262980906649372264101176,
1.46086480417294976792924673431, 3.19141236490110513849903019010, 4.64761748288521124520220416086, 5.76955345173946833740419292228, 7.17942763390166204355040609419, 7.925840375449905626544076446291, 8.385337755222518515279186404138, 10.11000696966439841569229177976, 10.66814260138686267558057944363, 11.50220190212957335422822887894
