| L(s) = 1 | + (−0.939 − 0.342i)2-s + (−1.21 − 1.23i)3-s + (0.766 + 0.642i)4-s + (−0.386 + 2.19i)5-s + (0.718 + 1.57i)6-s + (−0.766 + 0.642i)7-s + (−0.500 − 0.866i)8-s + (−0.0500 + 2.99i)9-s + (1.11 − 1.92i)10-s + (−1.14 − 6.48i)11-s + (−0.136 − 1.72i)12-s + (4.50 − 1.64i)13-s + (0.939 − 0.342i)14-s + (3.18 − 2.18i)15-s + (0.173 + 0.984i)16-s + (0.131 − 0.227i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (−0.701 − 0.712i)3-s + (0.383 + 0.321i)4-s + (−0.173 + 0.981i)5-s + (0.293 + 0.643i)6-s + (−0.289 + 0.242i)7-s + (−0.176 − 0.306i)8-s + (−0.0166 + 0.999i)9-s + (0.352 − 0.610i)10-s + (−0.344 − 1.95i)11-s + (−0.0394 − 0.498i)12-s + (1.24 − 0.454i)13-s + (0.251 − 0.0914i)14-s + (0.821 − 0.564i)15-s + (0.0434 + 0.246i)16-s + (0.0318 − 0.0551i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.392 + 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.306265 - 0.463611i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.306265 - 0.463611i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 + 0.342i)T \) |
| 3 | \( 1 + (1.21 + 1.23i)T \) |
| 7 | \( 1 + (0.766 - 0.642i)T \) |
| good | 5 | \( 1 + (0.386 - 2.19i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (1.14 + 6.48i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-4.50 + 1.64i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.131 + 0.227i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.10 + 3.65i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.07 + 1.74i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (9.17 + 3.34i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (1.87 + 1.56i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-5.70 + 9.88i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.52 + 1.64i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.493 - 2.80i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-2.04 + 1.71i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 6.11T + 53T^{2} \) |
| 59 | \( 1 + (1.25 - 7.09i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-6.89 + 5.78i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (5.76 - 2.10i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-5.31 + 9.20i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.29 - 3.97i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (9.89 + 3.60i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-14.0 - 5.11i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (5.59 + 9.69i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.77 - 10.0i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.96199895342726150189168032645, −10.71674431843330894460192164999, −9.146737700037923794231106584839, −8.205896000900851671731110317011, −7.38230828393602411364042483804, −6.20585946144234847042533222328, −5.80969425573659643321186654330, −3.61047101966930739270422650941, −2.47997241565166441741318974595, −0.52463231256893824967218092883,
1.51912000801930259636339377004, 3.90088554721868087419893742104, 4.79396687124637575238141706227, 5.88870032833929698721742021637, 6.92380911834040621871889998291, 8.044110899399510926201427606592, 9.132257432750511118684588254627, 9.732745079757949467731738081973, 10.56185374570822715797145376550, 11.49225255730628788435563919475
