| L(s) = 1 | + (0.781 − 0.623i)2-s + (1.60 − 0.628i)3-s + (0.222 − 0.974i)4-s + (0.651 + 2.13i)5-s + (0.859 − 1.48i)6-s + (0.690 − 0.398i)7-s + (−0.433 − 0.900i)8-s + (−0.0321 + 0.0298i)9-s + (1.84 + 1.26i)10-s + (−0.931 − 4.07i)11-s + (−0.256 − 1.70i)12-s + (1.67 − 2.45i)13-s + (0.291 − 0.742i)14-s + (2.38 + 3.01i)15-s + (−0.900 − 0.433i)16-s + (2.43 − 0.182i)17-s + ⋯ |
| L(s) = 1 | + (0.552 − 0.440i)2-s + (0.924 − 0.362i)3-s + (0.111 − 0.487i)4-s + (0.291 + 0.956i)5-s + (0.350 − 0.607i)6-s + (0.261 − 0.150i)7-s + (−0.153 − 0.318i)8-s + (−0.0107 + 0.00995i)9-s + (0.582 + 0.400i)10-s + (−0.280 − 1.23i)11-s + (−0.0739 − 0.490i)12-s + (0.463 − 0.679i)13-s + (0.0778 − 0.198i)14-s + (0.616 + 0.778i)15-s + (−0.225 − 0.108i)16-s + (0.590 − 0.0442i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.37078 - 0.944204i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.37078 - 0.944204i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.781 + 0.623i)T \) |
| 5 | \( 1 + (-0.651 - 2.13i)T \) |
| 43 | \( 1 + (4.50 - 4.76i)T \) |
| good | 3 | \( 1 + (-1.60 + 0.628i)T + (2.19 - 2.04i)T^{2} \) |
| 7 | \( 1 + (-0.690 + 0.398i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.931 + 4.07i)T + (-9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (-1.67 + 2.45i)T + (-4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-2.43 + 0.182i)T + (16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (-4.32 - 4.01i)T + (1.41 + 18.9i)T^{2} \) |
| 23 | \( 1 + (2.04 - 6.62i)T + (-19.0 - 12.9i)T^{2} \) |
| 29 | \( 1 + (-3.73 + 9.51i)T + (-21.2 - 19.7i)T^{2} \) |
| 31 | \( 1 + (10.0 - 1.51i)T + (29.6 - 9.13i)T^{2} \) |
| 37 | \( 1 + (-3.17 - 1.83i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.82 + 3.54i)T + (-9.12 + 39.9i)T^{2} \) |
| 47 | \( 1 + (10.9 + 2.49i)T + (42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (-2.19 - 3.21i)T + (-19.3 + 49.3i)T^{2} \) |
| 59 | \( 1 + (-2.16 - 1.04i)T + (36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (12.4 + 1.87i)T + (58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (-1.14 + 1.23i)T + (-5.00 - 66.8i)T^{2} \) |
| 71 | \( 1 + (4.61 - 1.42i)T + (58.6 - 39.9i)T^{2} \) |
| 73 | \( 1 + (-3.57 + 5.24i)T + (-26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (-1.99 - 3.45i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-11.7 + 4.62i)T + (60.8 - 56.4i)T^{2} \) |
| 89 | \( 1 + (0.549 + 1.39i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (-10.8 + 2.47i)T + (87.3 - 42.0i)T^{2} \) |
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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
−11.14409156613191343618641441374, −10.26694833834149485105374227931, −9.394785149334804916084782028162, −8.032995707177346331877624884775, −7.63640629981748829737036746423, −6.10568664832667070003220473353, −5.45429060102973154359367633745, −3.46934737478321148836078845146, −3.13722986326918731615741061889, −1.70578177738729528475032567429,
1.96926852326643837740067401150, 3.37418540113803424320745810447, 4.56440629311609272853557086854, 5.23766307532549966746684323251, 6.58784505291599005550485647987, 7.68700650285095485345819596354, 8.635135567668491126458164896641, 9.200307696592666303174975265097, 10.09146853051594881190084901162, 11.50396258184187279164594764949
