| L(s) = 1 | + (−0.150 + 1.40i)2-s + (−0.283 + 0.246i)3-s + (−1.95 − 0.423i)4-s + (2.45 − 0.352i)5-s + (−0.303 − 0.436i)6-s + (−1.80 + 3.94i)7-s + (0.890 − 2.68i)8-s + (−0.406 + 2.82i)9-s + (0.126 + 3.50i)10-s + (−0.456 + 1.55i)11-s + (0.659 − 0.360i)12-s + (1.85 − 0.845i)13-s + (−5.27 − 3.13i)14-s + (−0.609 + 0.703i)15-s + (3.64 + 1.65i)16-s + (−1.97 − 1.26i)17-s + ⋯ |
| L(s) = 1 | + (−0.106 + 0.994i)2-s + (−0.163 + 0.142i)3-s + (−0.977 − 0.211i)4-s + (1.09 − 0.157i)5-s + (−0.123 − 0.178i)6-s + (−0.681 + 1.49i)7-s + (0.314 − 0.949i)8-s + (−0.135 + 0.943i)9-s + (0.0399 + 1.10i)10-s + (−0.137 + 0.468i)11-s + (0.190 − 0.104i)12-s + (0.513 − 0.234i)13-s + (−1.41 − 0.836i)14-s + (−0.157 + 0.181i)15-s + (0.910 + 0.414i)16-s + (−0.478 − 0.307i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.543 - 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.543 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.490586 + 0.902563i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.490586 + 0.902563i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.150 - 1.40i)T \) |
| 23 | \( 1 + (-4.47 - 1.71i)T \) |
| good | 3 | \( 1 + (0.283 - 0.246i)T + (0.426 - 2.96i)T^{2} \) |
| 5 | \( 1 + (-2.45 + 0.352i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (1.80 - 3.94i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (0.456 - 1.55i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.85 + 0.845i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (1.97 + 1.26i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (1.17 + 1.82i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-2.85 + 4.43i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-3.79 + 4.37i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-5.95 - 0.856i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.0399 - 0.277i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-3.34 + 2.89i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 1.70T + 47T^{2} \) |
| 53 | \( 1 + (-11.4 - 5.23i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (10.7 - 4.91i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (10.0 + 8.70i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (4.29 + 14.6i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-2.37 + 0.697i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-8.91 + 5.73i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-1.23 - 2.70i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (1.47 + 0.212i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-5.94 - 6.86i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.71 - 11.9i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−13.35742725854740367682214114805, −12.25006353525331428146856762457, −10.73137775167935241270287994902, −9.557127842945182540454678938554, −9.073790845790075935069813446462, −7.894957603575217244023756805749, −6.38264324358611669483996667621, −5.72121390617652035604198823678, −4.78094999611786380432771346515, −2.46964872278843051819540583708,
1.09677266615861753964116863141, 3.05828595938627206968186552673, 4.23425021778690417815937408595, 5.94999764466406069938012814779, 6.92585852350300276094074172784, 8.608760225090879628549726301559, 9.570373174870180148307486751271, 10.40395010691696535806160182018, 11.03982234845011064218528961219, 12.38612716907121868204166487307
