| L(s) = 1 | + (−1.44 − 0.218i)2-s + (0.807 − 0.550i)3-s + (0.142 + 0.0440i)4-s + (0.0688 + 0.919i)5-s + (−1.29 + 0.621i)6-s + (2.44 + 1.17i)8-s + (−0.746 + 1.90i)9-s + (0.100 − 1.34i)10-s + (−0.0755 − 0.192i)11-s + (0.139 − 0.0430i)12-s + (−1.07 + 1.35i)13-s + (0.561 + 0.704i)15-s + (−3.53 − 2.40i)16-s + (3.76 + 3.49i)17-s + (1.49 − 2.59i)18-s + (1.62 + 2.81i)19-s + ⋯ |
| L(s) = 1 | + (−1.02 − 0.154i)2-s + (0.466 − 0.317i)3-s + (0.0714 + 0.0220i)4-s + (0.0308 + 0.411i)5-s + (−0.527 + 0.253i)6-s + (0.864 + 0.416i)8-s + (−0.248 + 0.634i)9-s + (0.0319 − 0.426i)10-s + (−0.0227 − 0.0580i)11-s + (0.0403 − 0.0124i)12-s + (−0.299 + 0.375i)13-s + (0.145 + 0.181i)15-s + (−0.883 − 0.602i)16-s + (0.912 + 0.846i)17-s + (0.353 − 0.611i)18-s + (0.372 + 0.646i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.768637 + 0.246118i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.768637 + 0.246118i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + (1.44 + 0.218i)T + (1.91 + 0.589i)T^{2} \) |
| 3 | \( 1 + (-0.807 + 0.550i)T + (1.09 - 2.79i)T^{2} \) |
| 5 | \( 1 + (-0.0688 - 0.919i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (0.0755 + 0.192i)T + (-8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (1.07 - 1.35i)T + (-2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-3.76 - 3.49i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.62 - 2.81i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.785 + 0.728i)T + (1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-1.32 + 5.80i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (4.19 - 7.26i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.83 + 2.41i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (-10.9 - 5.28i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (8.14 - 3.92i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (-5.30 - 0.800i)T + (44.9 + 13.8i)T^{2} \) |
| 53 | \( 1 + (0.791 + 0.244i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.368 + 4.91i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (8.44 - 2.60i)T + (50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (2.91 - 5.04i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.33 + 10.2i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (9.23 - 1.39i)T + (69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (2.27 + 3.94i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (10.0 + 12.5i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (2.76 - 7.05i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + 5.84T + 97T^{2} \) |
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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
−11.26235665694029531403217430522, −10.51892845939740681527286468390, −9.741822284706555472530494776425, −8.767575057586506011722410799987, −7.959773525873437727790359497967, −7.32694580418595429009707692684, −5.88610119889549135699441552071, −4.55200205212648920566658773630, −2.90078112425359606357595001389, −1.53497983031199227177708960996,
0.844234867888643360719770452643, 2.95132522296343643097052847831, 4.30672330641843052809934712794, 5.49848102371963493525680561683, 7.05362282140161049143988278332, 7.84161897219302332090136731854, 8.904757239700548506666746857229, 9.333974730410419411277068192448, 10.11423901449277852737375915578, 11.20449863127546389423013087859
