| L(s) = 1 | + (−0.395 − 1.96i)2-s + (−1.67 + 0.448i)3-s + (−3.68 + 1.55i)4-s + (−1.60 − 0.428i)5-s + (1.54 + 3.10i)6-s + (6.62 − 2.26i)7-s + (4.49 + 6.61i)8-s + (2.59 − 1.50i)9-s + (−0.208 + 3.30i)10-s + (3.06 − 0.822i)11-s + (5.47 − 4.24i)12-s + (0.428 + 0.428i)13-s + (−7.06 − 12.0i)14-s + 2.87·15-s + (11.1 − 11.4i)16-s + (1.13 + 0.655i)17-s + ⋯ |
| L(s) = 1 | + (−0.197 − 0.980i)2-s + (−0.557 + 0.149i)3-s + (−0.921 + 0.387i)4-s + (−0.320 − 0.0857i)5-s + (0.256 + 0.517i)6-s + (0.945 − 0.324i)7-s + (0.562 + 0.827i)8-s + (0.288 − 0.166i)9-s + (−0.0208 + 0.330i)10-s + (0.278 − 0.0747i)11-s + (0.456 − 0.353i)12-s + (0.0329 + 0.0329i)13-s + (−0.504 − 0.863i)14-s + 0.191·15-s + (0.699 − 0.714i)16-s + (0.0668 + 0.0385i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 + 0.563i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.826 + 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.262799 - 0.852061i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.262799 - 0.852061i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.395 + 1.96i)T \) |
| 3 | \( 1 + (1.67 - 0.448i)T \) |
| 7 | \( 1 + (-6.62 + 2.26i)T \) |
| good | 5 | \( 1 + (1.60 + 0.428i)T + (21.6 + 12.5i)T^{2} \) |
| 11 | \( 1 + (-3.06 + 0.822i)T + (104. - 60.5i)T^{2} \) |
| 13 | \( 1 + (-0.428 - 0.428i)T + 169iT^{2} \) |
| 17 | \( 1 + (-1.13 - 0.655i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-7.33 + 27.3i)T + (-312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (35.7 - 20.6i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-33.9 + 33.9i)T - 841iT^{2} \) |
| 31 | \( 1 + (29.9 + 17.2i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-18.3 + 68.3i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + 63.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-31.4 - 31.4i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-15.2 + 8.78i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-3.38 + 0.906i)T + (2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (-3.69 - 13.7i)T + (-3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (-13.7 + 51.1i)T + (-3.22e3 - 1.86e3i)T^{2} \) |
| 67 | \( 1 + (-1.91 - 7.13i)T + (-3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 - 44.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-68.1 + 118. i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-38.8 - 67.2i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (61.7 + 61.7i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (52.2 + 90.4i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 34.8iT - 9.40e3T^{2} \) |
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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
−11.25066385007021137893616063596, −10.18353796379001960775915307491, −9.347193462980832951334110977552, −8.213830240435415219157715709401, −7.40897247252232163965503625635, −5.76476920041320723779389877936, −4.61501997694334121231125590667, −3.83717733016625563103897436632, −2.06881834439736234817965217098, −0.52219633695969252267366241351,
1.47165241873218948595009283582, 3.90788724142933919404952030452, 5.02302989786700317240732821926, 5.89182966378203192048796408943, 6.90984624203537681813137874709, 7.962392585462336412204384892253, 8.514749767820786858921775555381, 9.862747720404051489751223671675, 10.63849239122609877645440008660, 11.88115833143328228888856373950
