| L(s) = 1 | + (−0.988 + 0.149i)2-s + (1.69 + 0.347i)3-s + (0.955 − 0.294i)4-s + (−2.92 − 1.99i)5-s + (−1.72 − 0.0909i)6-s + (−0.149 + 2.64i)7-s + (−0.900 + 0.433i)8-s + (2.75 + 1.18i)9-s + (3.19 + 1.53i)10-s + (−3.39 + 0.512i)11-s + (1.72 − 0.167i)12-s + (1.14 − 2.91i)13-s + (−0.245 − 2.63i)14-s + (−4.27 − 4.40i)15-s + (0.826 − 0.563i)16-s + (−1.75 − 7.70i)17-s + ⋯ |
| L(s) = 1 | + (−0.699 + 0.105i)2-s + (0.979 + 0.200i)3-s + (0.477 − 0.147i)4-s + (−1.30 − 0.892i)5-s + (−0.706 − 0.0371i)6-s + (−0.0566 + 0.998i)7-s + (−0.318 + 0.153i)8-s + (0.919 + 0.393i)9-s + (1.00 + 0.486i)10-s + (−1.02 + 0.154i)11-s + (0.497 − 0.0484i)12-s + (0.317 − 0.808i)13-s + (−0.0656 − 0.704i)14-s + (−1.10 − 1.13i)15-s + (0.206 − 0.140i)16-s + (−0.426 − 1.86i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.995132 - 0.504200i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.995132 - 0.504200i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.988 - 0.149i)T \) |
| 3 | \( 1 + (-1.69 - 0.347i)T \) |
| 7 | \( 1 + (0.149 - 2.64i)T \) |
| good | 5 | \( 1 + (2.92 + 1.99i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (3.39 - 0.512i)T + (10.5 - 3.24i)T^{2} \) |
| 13 | \( 1 + (-1.14 + 2.91i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (1.75 + 7.70i)T + (-15.3 + 7.37i)T^{2} \) |
| 19 | \( 1 - 5.49T + 19T^{2} \) |
| 23 | \( 1 + (-6.29 + 1.94i)T + (19.0 - 12.9i)T^{2} \) |
| 29 | \( 1 + (-7.78 - 2.39i)T + (23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (3.32 + 5.76i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.02 + 8.85i)T + (-33.3 + 16.0i)T^{2} \) |
| 41 | \( 1 + (2.10 + 1.43i)T + (14.9 + 38.1i)T^{2} \) |
| 43 | \( 1 + (-3.82 + 2.60i)T + (15.7 - 40.0i)T^{2} \) |
| 47 | \( 1 + (1.37 - 0.207i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (-1.10 + 4.84i)T + (-47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 + (-0.200 - 2.68i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (12.3 + 3.79i)T + (50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (-5.83 - 10.0i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.565 - 2.47i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-4.69 + 5.88i)T + (-16.2 - 71.1i)T^{2} \) |
| 79 | \( 1 + (-1.05 + 1.82i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.432 + 1.10i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (0.740 - 0.928i)T + (-19.8 - 86.7i)T^{2} \) |
| 97 | \( 1 + (-2.40 + 4.16i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.642126936078558105828948993789, −8.984078951957525840495307424827, −8.466107941654095258355387965928, −7.64854107905383123003646629371, −7.18274064430145073538101404450, −5.34875023372823830696680446753, −4.77963517640804503075439726030, −3.28316371661915727745915446503, −2.55155198743716124410046901665, −0.66791081174933220338725094175,
1.33072538992757206401686008577, 2.97729926442009239724392271824, 3.52974912419352226456586401210, 4.54863880506814917751484538211, 6.57295463761002361828550277510, 7.07661782904414992769107195381, 7.929213718049520608696964211685, 8.269917382276721641702658144293, 9.353324623760137275102771981605, 10.47911957539672350195763687631
