| L(s) = 1 | + (−0.766 + 0.642i)2-s + (0.164 + 1.72i)3-s + (0.173 − 0.984i)4-s + (0.421 − 1.15i)5-s + (−1.23 − 1.21i)6-s + (−0.0757 − 0.131i)7-s + (0.500 + 0.866i)8-s + (−2.94 + 0.565i)9-s + (0.421 + 1.15i)10-s + 4.18i·11-s + (1.72 + 0.137i)12-s + (1.93 + 5.32i)13-s + (0.142 + 0.0518i)14-s + (2.06 + 0.536i)15-s + (−0.939 − 0.342i)16-s + (−1.59 + 4.38i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.454i)2-s + (0.0946 + 0.995i)3-s + (0.0868 − 0.492i)4-s + (0.188 − 0.517i)5-s + (−0.503 − 0.496i)6-s + (−0.0286 − 0.0496i)7-s + (0.176 + 0.306i)8-s + (−0.982 + 0.188i)9-s + (0.133 + 0.366i)10-s + 1.26i·11-s + (0.498 + 0.0398i)12-s + (0.537 + 1.47i)13-s + (0.0380 + 0.0138i)14-s + (0.533 + 0.138i)15-s + (−0.234 − 0.0855i)16-s + (−0.386 + 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.429732 + 0.849965i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.429732 + 0.849965i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 3 | \( 1 + (-0.164 - 1.72i)T \) |
| 19 | \( 1 + (-2.66 + 3.45i)T \) |
| good | 5 | \( 1 + (-0.421 + 1.15i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.0757 + 0.131i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 4.18iT - 11T^{2} \) |
| 13 | \( 1 + (-1.93 - 5.32i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (1.59 - 4.38i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (6.17 + 1.08i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (1.50 - 8.53i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + 9.03iT - 31T^{2} \) |
| 37 | \( 1 - 3.92iT - 37T^{2} \) |
| 41 | \( 1 + (-9.15 + 7.68i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.09 - 6.19i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (3.11 + 0.549i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (4.00 + 3.35i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-1.20 - 6.84i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (1.07 - 0.389i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-7.10 + 8.47i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.44 + 1.21i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-1.66 - 9.44i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.210 + 0.578i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-10.0 + 5.79i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.130 + 0.738i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (4.50 + 5.37i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−11.56010101934449301696217180004, −10.74998670602211575373762439892, −9.685084405729040677772796090550, −9.214039306310523163658671613929, −8.370486408004513600144183396955, −7.12232906145912465461447133920, −6.02486670339950113432761235758, −4.83122354030266396873224867877, −4.01331761840812787753620745149, −1.97277553821107792160356247163,
0.804372237694028475994534544999, 2.55580323551903476916277540983, 3.43715170858244437908830503242, 5.63019977801151286627599472284, 6.40396473980790479340934571161, 7.71201171681415999728394775610, 8.197472836186775760575954426955, 9.293964159540946137452583512282, 10.44749012504032369444538951524, 11.18472129328683967634952787552
