| L(s) = 1 | + (0.519 + 1.31i)2-s + (−1.95 − 0.662i)3-s + (−1.46 + 1.36i)4-s + (0.253 − 0.513i)5-s + (−0.141 − 2.91i)6-s + (2.58 + 0.541i)7-s + (−2.55 − 1.21i)8-s + (0.990 + 0.760i)9-s + (0.807 + 0.0665i)10-s + (4.32 + 3.79i)11-s + (3.75 − 1.69i)12-s + (−2.69 + 4.02i)13-s + (0.632 + 3.68i)14-s + (−0.835 + 0.835i)15-s + (0.268 − 3.99i)16-s + (−4.98 + 1.33i)17-s + ⋯ |
| L(s) = 1 | + (0.367 + 0.930i)2-s + (−1.12 − 0.382i)3-s + (−0.730 + 0.682i)4-s + (0.113 − 0.229i)5-s + (−0.0578 − 1.18i)6-s + (0.978 + 0.204i)7-s + (−0.903 − 0.428i)8-s + (0.330 + 0.253i)9-s + (0.255 + 0.0210i)10-s + (1.30 + 1.14i)11-s + (1.08 − 0.490i)12-s + (−0.746 + 1.11i)13-s + (0.168 + 0.985i)14-s + (−0.215 + 0.215i)15-s + (0.0670 − 0.997i)16-s + (−1.20 + 0.323i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 - 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.822 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.266504 + 0.853249i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.266504 + 0.853249i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.519 - 1.31i)T \) |
| 7 | \( 1 + (-2.58 - 0.541i)T \) |
| good | 3 | \( 1 + (1.95 + 0.662i)T + (2.38 + 1.82i)T^{2} \) |
| 5 | \( 1 + (-0.253 + 0.513i)T + (-3.04 - 3.96i)T^{2} \) |
| 11 | \( 1 + (-4.32 - 3.79i)T + (1.43 + 10.9i)T^{2} \) |
| 13 | \( 1 + (2.69 - 4.02i)T + (-4.97 - 12.0i)T^{2} \) |
| 17 | \( 1 + (4.98 - 1.33i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (6.58 + 0.431i)T + (18.8 + 2.47i)T^{2} \) |
| 23 | \( 1 + (-1.22 - 0.936i)T + (5.95 + 22.2i)T^{2} \) |
| 29 | \( 1 + (5.88 - 1.17i)T + (26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 + (-2.35 - 1.36i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.0778 - 0.157i)T + (-22.5 - 29.3i)T^{2} \) |
| 41 | \( 1 + (2.24 - 0.928i)T + (28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (-5.36 - 1.06i)T + (39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (-7.80 - 2.09i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (4.51 - 5.15i)T + (-6.91 - 52.5i)T^{2} \) |
| 59 | \( 1 + (-0.293 - 4.47i)T + (-58.4 + 7.70i)T^{2} \) |
| 61 | \( 1 + (5.10 + 5.82i)T + (-7.96 + 60.4i)T^{2} \) |
| 67 | \( 1 + (2.04 - 6.03i)T + (-53.1 - 40.7i)T^{2} \) |
| 71 | \( 1 + (-4.24 + 10.2i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (-2.38 - 0.314i)T + (70.5 + 18.8i)T^{2} \) |
| 79 | \( 1 + (4.38 + 1.17i)T + (68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (0.806 - 1.20i)T + (-31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (2.14 + 16.2i)T + (-85.9 + 23.0i)T^{2} \) |
| 97 | \( 1 - 19.6T + 97T^{2} \) |
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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
−11.68030949405973019798862936390, −10.87633219139534546742169180252, −9.240129858516433078406202160072, −8.830953748859255896223061492128, −7.33385164561168829605162321042, −6.76269652763661222007113727138, −5.95187088122237379604469872967, −4.71256257938616774904075212484, −4.35065195591230432730067314038, −1.81963037515359299356062448260,
0.58065471514712546195538248153, 2.38019783718134971973953377016, 3.99834623331611928139782464956, 4.80202666924646563316479079695, 5.75423778249468833428280345249, 6.57604514923512011035588662069, 8.285874141143083052229325716125, 9.112381449605551306510633371573, 10.42907916622314938696834109065, 10.84775847177173507754246602591
