| L(s) = 1 | + (0.939 − 0.342i)2-s + (−1.62 + 0.586i)3-s + (0.766 − 0.642i)4-s + (0.184 + 1.04i)5-s + (−1.33 + 1.10i)6-s + (0.766 + 0.642i)7-s + (0.500 − 0.866i)8-s + (2.31 − 1.91i)9-s + (0.531 + 0.920i)10-s + (−0.543 + 3.08i)11-s + (−0.871 + 1.49i)12-s + (2.26 + 0.825i)13-s + (0.939 + 0.342i)14-s + (−0.914 − 1.59i)15-s + (0.173 − 0.984i)16-s + (3.26 + 5.65i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (−0.940 + 0.338i)3-s + (0.383 − 0.321i)4-s + (0.0825 + 0.467i)5-s + (−0.543 + 0.452i)6-s + (0.289 + 0.242i)7-s + (0.176 − 0.306i)8-s + (0.770 − 0.636i)9-s + (0.167 + 0.290i)10-s + (−0.163 + 0.929i)11-s + (−0.251 + 0.432i)12-s + (0.629 + 0.229i)13-s + (0.251 + 0.0914i)14-s + (−0.235 − 0.412i)15-s + (0.0434 − 0.246i)16-s + (0.791 + 1.37i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.50090 + 0.487902i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50090 + 0.487902i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 + 0.342i)T \) |
| 3 | \( 1 + (1.62 - 0.586i)T \) |
| 7 | \( 1 + (-0.766 - 0.642i)T \) |
| good | 5 | \( 1 + (-0.184 - 1.04i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (0.543 - 3.08i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-2.26 - 0.825i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-3.26 - 5.65i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.751 - 1.30i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.03 + 1.70i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (0.635 - 0.231i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (1.62 - 1.36i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (2.73 + 4.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.25 - 0.456i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.48 + 8.43i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (6.38 + 5.35i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 2.45T + 53T^{2} \) |
| 59 | \( 1 + (-1.42 - 8.06i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (7.39 + 6.20i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (8.57 + 3.12i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-3.25 - 5.64i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.37 + 4.10i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.08 + 1.84i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (14.4 - 5.26i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-7.63 + 13.2i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.85 + 16.1i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.44366767410659790652635211609, −10.58647511123977756598784524457, −10.15847617864888383887780703264, −8.819011203227450480787587976825, −7.37512221570770992052625427421, −6.41810803110130525609128694056, −5.59001887700088565637757639818, −4.56232241862484540180981506730, −3.52375607690266678638906554433, −1.74381996119232987389011737514,
1.11360472357864550771968820125, 3.14788856640393144904302348838, 4.67766167726614287674250746945, 5.38174359817272990221360283246, 6.29345298336605234533575023817, 7.33283762953564940212883044667, 8.218118342626810790093459190544, 9.487028988910289835249591777939, 10.81311232406704821298577777524, 11.33089055790361856491781611526
