| L(s) = 1 | + (1.16 − 0.795i)2-s + (−1.63 + 0.575i)3-s + (0.734 − 1.86i)4-s + (−3.21 − 1.17i)5-s + (−1.45 + 1.97i)6-s + (−2.85 − 0.504i)7-s + (−0.621 − 2.75i)8-s + (2.33 − 1.87i)9-s + (−4.69 + 1.18i)10-s + (−0.927 − 2.54i)11-s + (−0.129 + 3.46i)12-s + (1.60 + 1.91i)13-s + (−3.74 + 1.68i)14-s + (5.92 + 0.0624i)15-s + (−2.92 − 2.73i)16-s + (4.30 − 2.48i)17-s + ⋯ |
| L(s) = 1 | + (0.826 − 0.562i)2-s + (−0.943 + 0.332i)3-s + (0.367 − 0.930i)4-s + (−1.43 − 0.523i)5-s + (−0.592 + 0.805i)6-s + (−1.08 − 0.190i)7-s + (−0.219 − 0.975i)8-s + (0.779 − 0.626i)9-s + (−1.48 + 0.376i)10-s + (−0.279 − 0.768i)11-s + (−0.0372 + 0.999i)12-s + (0.444 + 0.530i)13-s + (−1.00 + 0.450i)14-s + (1.53 + 0.0161i)15-s + (−0.730 − 0.682i)16-s + (1.04 − 0.603i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.886 + 0.462i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.886 + 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.177084 - 0.722138i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.177084 - 0.722138i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.16 + 0.795i)T \) |
| 3 | \( 1 + (1.63 - 0.575i)T \) |
| good | 5 | \( 1 + (3.21 + 1.17i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (2.85 + 0.504i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (0.927 + 2.54i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.60 - 1.91i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-4.30 + 2.48i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.31 - 2.27i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.157 - 0.895i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (7.00 + 5.87i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-2.90 + 0.512i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-9.78 + 5.64i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.04 + 3.62i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (6.12 - 2.23i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.225 - 1.28i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + (-2.96 + 8.13i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.720 - 0.127i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-5.00 + 4.19i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (5.18 + 8.97i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7.34 - 12.7i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.64 + 6.73i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-3.15 + 3.75i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (4.90 + 2.82i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.126 - 0.0461i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.75512788174064543069398823038, −11.39092495507917310305719388023, −10.25584487864981780431455244595, −9.316432316361639639954247403042, −7.67552853825191026541619379853, −6.42449857732406870320165899461, −5.45235808660147887819955920559, −4.13458668808941884197405127752, −3.48758454866539949206201681290, −0.53782555334042522358127765706,
3.12514811423593178951422734535, 4.23880611504575208070206918609, 5.55509813261348290286711432448, 6.64803599313090193569755516976, 7.34193480377778209233716897220, 8.258414432795648530435842964320, 10.12986714339362946819172005730, 11.17227610085269911404492960707, 11.97375864440990255759940350968, 12.70253075383167599176605620828
