| L(s) = 1 | + (0.415 + 0.909i)2-s + (−3.22 + 0.946i)3-s + (−0.654 + 0.755i)4-s + (−0.841 − 0.540i)5-s + (−2.20 − 2.53i)6-s + (−0.324 − 2.25i)7-s + (−0.959 − 0.281i)8-s + (6.97 − 4.48i)9-s + (0.142 − 0.989i)10-s + (−1.18 + 2.59i)11-s + (1.39 − 3.05i)12-s + (0.360 − 2.50i)13-s + (1.91 − 1.23i)14-s + (3.22 + 0.946i)15-s + (−0.142 − 0.989i)16-s + (−3.51 − 4.05i)17-s + ⋯ |
| L(s) = 1 | + (0.293 + 0.643i)2-s + (−1.86 + 0.546i)3-s + (−0.327 + 0.377i)4-s + (−0.376 − 0.241i)5-s + (−0.898 − 1.03i)6-s + (−0.122 − 0.853i)7-s + (−0.339 − 0.0996i)8-s + (2.32 − 1.49i)9-s + (0.0450 − 0.313i)10-s + (−0.356 + 0.781i)11-s + (0.402 − 0.882i)12-s + (0.0998 − 0.694i)13-s + (0.512 − 0.329i)14-s + (0.832 + 0.244i)15-s + (−0.0355 − 0.247i)16-s + (−0.852 − 0.983i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.258117 - 0.185614i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.258117 - 0.185614i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.415 - 0.909i)T \) |
| 5 | \( 1 + (0.841 + 0.540i)T \) |
| 23 | \( 1 + (4.47 + 1.71i)T \) |
| good | 3 | \( 1 + (3.22 - 0.946i)T + (2.52 - 1.62i)T^{2} \) |
| 7 | \( 1 + (0.324 + 2.25i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (1.18 - 2.59i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.360 + 2.50i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (3.51 + 4.05i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-2.49 + 2.87i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (1.47 + 1.70i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (7.80 + 2.29i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (-0.242 + 0.155i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (5.33 + 3.42i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (6.39 - 1.87i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 + (-1.60 - 11.1i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-0.457 + 3.17i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (0.415 + 0.121i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (1.33 + 2.92i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (-0.758 - 1.66i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-8.00 + 9.23i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (-0.821 + 5.71i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (7.75 - 4.98i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (16.8 - 4.95i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (4.32 + 2.77i)T + (40.2 + 88.2i)T^{2} \) |
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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
−12.01036721192648605744630576843, −11.09942032367732087704282748826, −10.27978025062445649279602075368, −9.320902102951159212695707063784, −7.49230619108651486262624315211, −6.88811985557970107472444911290, −5.65735518702934014361457623729, −4.80453547208295112164885473487, −3.98064307675283738196477708695, −0.29648738018086831334015074345,
1.81369442236850725504447982402, 3.93965700809520093287302646213, 5.35480694976579928678365562422, 5.98182346923692724758996033638, 7.02758304217902208245989696771, 8.464211608720253332432347637075, 9.976769228284715125974861908731, 10.92545789825283354726266424578, 11.48904483040140113361446356917, 12.19479717617633827604694193890
