| L(s) = 1 | + (0.866 + 1.5i)3-s + (−1.73 + 1.73i)5-s + (8.96 − 2.40i)7-s + (3 − 5.19i)9-s + (−1.96 + 7.33i)11-s + (3.92 + 12.3i)13-s + (−4.09 − 1.09i)15-s + (14.3 + 8.25i)17-s + (4.10 + 15.3i)19-s + (11.3 + 11.3i)21-s + (−21.1 + 12.2i)23-s + 19i·25-s + 25.9·27-s + (−27.3 − 47.3i)29-s + (40.6 − 40.6i)31-s + ⋯ |
| L(s) = 1 | + (0.288 + 0.5i)3-s + (−0.346 + 0.346i)5-s + (1.28 − 0.343i)7-s + (0.333 − 0.577i)9-s + (−0.178 + 0.666i)11-s + (0.302 + 0.953i)13-s + (−0.273 − 0.0732i)15-s + (0.841 + 0.485i)17-s + (0.216 + 0.806i)19-s + (0.541 + 0.541i)21-s + (−0.921 + 0.531i)23-s + 0.760i·25-s + 0.962·27-s + (−0.943 − 1.63i)29-s + (1.31 − 1.31i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.693 - 0.720i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.693 - 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.68363 + 0.716573i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.68363 + 0.716573i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + (-3.92 - 12.3i)T \) |
| good | 3 | \( 1 + (-0.866 - 1.5i)T + (-4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 + (1.73 - 1.73i)T - 25iT^{2} \) |
| 7 | \( 1 + (-8.96 + 2.40i)T + (42.4 - 24.5i)T^{2} \) |
| 11 | \( 1 + (1.96 - 7.33i)T + (-104. - 60.5i)T^{2} \) |
| 17 | \( 1 + (-14.3 - 8.25i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-4.10 - 15.3i)T + (-312. + 180.5i)T^{2} \) |
| 23 | \( 1 + (21.1 - 12.2i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (27.3 + 47.3i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-40.6 + 40.6i)T - 961iT^{2} \) |
| 37 | \( 1 + (1.25 - 4.69i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + (-11.2 - 3.01i)T + (1.45e3 + 840.5i)T^{2} \) |
| 43 | \( 1 + (-3.77 - 2.17i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (33 + 33i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + 3.89T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-7.16 + 1.91i)T + (3.01e3 - 1.74e3i)T^{2} \) |
| 61 | \( 1 + (-24.6 + 42.7i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (30.1 + 8.08i)T + (3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 + (13.9 + 52.1i)T + (-4.36e3 + 2.52e3i)T^{2} \) |
| 73 | \( 1 + (80.7 + 80.7i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 110.T + 6.24e3T^{2} \) |
| 83 | \( 1 + (8.16 - 8.16i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (-22.2 + 82.9i)T + (-6.85e3 - 3.96e3i)T^{2} \) |
| 97 | \( 1 + (-8.51 - 31.7i)T + (-8.14e3 + 4.70e3i)T^{2} \) |
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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
−11.94993244964977399013652316369, −11.46163040911984462467896989967, −10.19849975491419305656450875772, −9.522984257184055634414501825215, −8.122184612154136433790084915623, −7.47449130257700872860388527700, −6.02573145051311656499904669677, −4.48326078650589942982932582054, −3.73905818295395047995051184625, −1.71728713673873130648498519277,
1.22608529742631542071344353920, 2.85456173105169993781868189169, 4.65402973869674115628072018058, 5.56022089639424989454668028766, 7.20990287904042332734472016431, 8.182189100420800587998404974768, 8.572194325830322445133516485081, 10.24950055381666025026753896700, 11.12704190103729128074729081993, 12.10197455589481498589196891026
