| L(s) = 1 | + (0.422 + 0.906i)2-s + (−1.65 + 0.510i)3-s + (−0.642 + 0.766i)4-s + (2.18 − 0.473i)5-s + (−1.16 − 1.28i)6-s + (0.689 + 2.57i)7-s + (−0.965 − 0.258i)8-s + (2.47 − 1.68i)9-s + (1.35 + 1.78i)10-s + (1.23 − 0.715i)11-s + (0.672 − 1.59i)12-s + (1.91 − 1.33i)13-s + (−2.04 + 1.71i)14-s + (−3.37 + 1.89i)15-s + (−0.173 − 0.984i)16-s + (2.73 + 5.87i)17-s + ⋯ |
| L(s) = 1 | + (0.298 + 0.640i)2-s + (−0.955 + 0.294i)3-s + (−0.321 + 0.383i)4-s + (0.977 − 0.211i)5-s + (−0.474 − 0.524i)6-s + (0.260 + 0.972i)7-s + (−0.341 − 0.0915i)8-s + (0.826 − 0.563i)9-s + (0.427 + 0.563i)10-s + (0.373 − 0.215i)11-s + (0.194 − 0.460i)12-s + (0.530 − 0.371i)13-s + (−0.545 + 0.457i)14-s + (−0.871 + 0.490i)15-s + (−0.0434 − 0.246i)16-s + (0.663 + 1.42i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.233 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.233 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.920177 + 1.16712i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.920177 + 1.16712i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.422 - 0.906i)T \) |
| 3 | \( 1 + (1.65 - 0.510i)T \) |
| 5 | \( 1 + (-2.18 + 0.473i)T \) |
| 19 | \( 1 + (4.35 - 0.234i)T \) |
| good | 7 | \( 1 + (-0.689 - 2.57i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-1.23 + 0.715i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.91 + 1.33i)T + (4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (-2.73 - 5.87i)T + (-10.9 + 13.0i)T^{2} \) |
| 23 | \( 1 + (-5.00 + 0.437i)T + (22.6 - 3.99i)T^{2} \) |
| 29 | \( 1 + (5.61 - 2.04i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (4.43 - 7.67i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.456 - 0.456i)T - 37iT^{2} \) |
| 41 | \( 1 + (-1.84 + 0.326i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (7.64 + 0.668i)T + (42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (-4.44 - 2.07i)T + (30.2 + 36.0i)T^{2} \) |
| 53 | \( 1 + (-6.42 + 0.562i)T + (52.1 - 9.20i)T^{2} \) |
| 59 | \( 1 + (-2.68 - 0.975i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (1.15 + 0.970i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (2.84 - 6.10i)T + (-43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (7.59 + 9.05i)T + (-12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (2.62 - 3.75i)T + (-24.9 - 68.5i)T^{2} \) |
| 79 | \( 1 + (-9.58 + 1.68i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (2.37 + 8.85i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-2.49 + 14.1i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-9.66 + 4.50i)T + (62.3 - 74.3i)T^{2} \) |
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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
−10.86991060521630529677778049890, −10.24682120672333677348103600620, −8.996348467965430546723081155370, −8.605943008479034294196120364747, −7.07910000283074520563023009101, −6.04426463594188476853334881873, −5.70915347853335218051465922553, −4.81789612199569139992057976703, −3.50354501699601811838126360005, −1.62264172795194291179723848223,
0.975951904617785928428998986528, 2.17941309475944561834923416375, 3.86033504215243896469003245342, 4.90009157855958765066546807094, 5.77701269577494496301465404494, 6.75022507101381990159231562777, 7.49319203974483559343884180561, 9.122449603133409819683791840224, 9.858597693379199972190998272820, 10.71963421134600298350615702756
