| L(s) = 1 | + (−1.65 + 1.23i)2-s + (−0.571 + 1.63i)3-s + (0.648 − 2.16i)4-s + (−0.0690 − 1.18i)5-s + (−1.06 − 3.41i)6-s + (−1.59 + 2.11i)7-s + (0.185 + 0.509i)8-s + (−2.34 − 1.86i)9-s + (1.57 + 1.87i)10-s + (−2.44 + 3.71i)11-s + (3.17 + 2.30i)12-s + (0.158 + 0.117i)13-s + (0.0381 − 5.46i)14-s + (1.97 + 0.564i)15-s + (2.84 + 1.87i)16-s + (−0.541 − 3.06i)17-s + ⋯ |
| L(s) = 1 | + (−1.17 + 0.871i)2-s + (−0.330 + 0.943i)3-s + (0.324 − 1.08i)4-s + (−0.0308 − 0.530i)5-s + (−0.436 − 1.39i)6-s + (−0.602 + 0.797i)7-s + (0.0655 + 0.180i)8-s + (−0.782 − 0.623i)9-s + (0.498 + 0.594i)10-s + (−0.736 + 1.11i)11-s + (0.915 + 0.664i)12-s + (0.0438 + 0.0326i)13-s + (0.0101 − 1.45i)14-s + (0.510 + 0.145i)15-s + (0.711 + 0.467i)16-s + (−0.131 − 0.744i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.193697 - 0.0423689i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.193697 - 0.0423689i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.571 - 1.63i)T \) |
| 7 | \( 1 + (1.59 - 2.11i)T \) |
| good | 2 | \( 1 + (1.65 - 1.23i)T + (0.573 - 1.91i)T^{2} \) |
| 5 | \( 1 + (0.0690 + 1.18i)T + (-4.96 + 0.580i)T^{2} \) |
| 11 | \( 1 + (2.44 - 3.71i)T + (-4.35 - 10.1i)T^{2} \) |
| 13 | \( 1 + (-0.158 - 0.117i)T + (3.72 + 12.4i)T^{2} \) |
| 17 | \( 1 + (0.541 + 3.06i)T + (-15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (2.54 + 0.448i)T + (17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (2.00 + 8.44i)T + (-20.5 + 10.3i)T^{2} \) |
| 29 | \( 1 + (1.41 + 0.611i)T + (19.9 + 21.0i)T^{2} \) |
| 31 | \( 1 + (-0.462 - 1.94i)T + (-27.7 + 13.9i)T^{2} \) |
| 37 | \( 1 + (-3.03 + 2.54i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (-1.70 - 0.198i)T + (39.8 + 9.45i)T^{2} \) |
| 43 | \( 1 + (-4.13 - 2.72i)T + (17.0 + 39.4i)T^{2} \) |
| 47 | \( 1 + (5.32 + 1.26i)T + (42.0 + 21.0i)T^{2} \) |
| 53 | \( 1 + 10.1iT - 53T^{2} \) |
| 59 | \( 1 + (-8.98 - 4.51i)T + (35.2 + 47.3i)T^{2} \) |
| 61 | \( 1 + (-0.212 + 0.0637i)T + (50.9 - 33.5i)T^{2} \) |
| 67 | \( 1 + (0.212 + 0.0248i)T + (65.1 + 15.4i)T^{2} \) |
| 71 | \( 1 + (4.19 - 11.5i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-2.17 - 0.383i)T + (68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (2.40 + 3.22i)T + (-22.6 + 75.6i)T^{2} \) |
| 83 | \( 1 + (2.12 - 0.248i)T + (80.7 - 19.1i)T^{2} \) |
| 89 | \( 1 + (9.38 + 7.87i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (7.09 + 14.1i)T + (-57.9 + 77.8i)T^{2} \) |
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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
−10.11596982907527193727720133325, −9.801366510634197436635559742621, −8.810299687270394202010747847819, −8.428830131265580391025299365233, −7.10427839860607223014562502323, −6.26970362010912235309379079517, −5.27387724456208121946769326577, −4.32684201801930262686006687651, −2.62786332169467697997115629830, −0.19200958035643800639800030209,
1.18124458233409891419683381990, 2.56985536114119261874355565223, 3.54967338556935344193809998657, 5.56813611980105974070085446447, 6.45930416103345358267470345352, 7.56872291132727917011310228802, 8.099081298037260242052444997283, 9.118034399248050265190217540408, 10.17176281894352690927943338541, 10.89040880991463570212665419360
