| L(s) = 1 | + (−1.37 + 0.322i)2-s + (1.59 − 0.667i)3-s + (1.79 − 0.889i)4-s + (−1.27 − 3.49i)5-s + (−1.98 + 1.43i)6-s + (−1.63 + 0.287i)7-s + (−2.17 + 1.80i)8-s + (2.10 − 2.13i)9-s + (2.87 + 4.39i)10-s + (0.526 + 0.191i)11-s + (2.26 − 2.61i)12-s + (1.99 + 1.66i)13-s + (2.15 − 0.922i)14-s + (−4.36 − 4.73i)15-s + (2.41 − 3.18i)16-s + (4.50 + 2.60i)17-s + ⋯ |
| L(s) = 1 | + (−0.973 + 0.228i)2-s + (0.922 − 0.385i)3-s + (0.895 − 0.444i)4-s + (−0.568 − 1.56i)5-s + (−0.810 + 0.585i)6-s + (−0.616 + 0.108i)7-s + (−0.770 + 0.637i)8-s + (0.702 − 0.711i)9-s + (0.910 + 1.39i)10-s + (0.158 + 0.0577i)11-s + (0.655 − 0.755i)12-s + (0.551 + 0.463i)13-s + (0.575 − 0.246i)14-s + (−1.12 − 1.22i)15-s + (0.604 − 0.796i)16-s + (1.09 + 0.631i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.740184 - 0.363114i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.740184 - 0.363114i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.37 - 0.322i)T \) |
| 3 | \( 1 + (-1.59 + 0.667i)T \) |
| good | 5 | \( 1 + (1.27 + 3.49i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (1.63 - 0.287i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.526 - 0.191i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.99 - 1.66i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-4.50 - 2.60i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.925 + 0.534i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.54 - 8.77i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (0.702 + 0.837i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (3.17 + 0.559i)T + (29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.17 + 5.50i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.556 - 0.662i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (0.690 - 1.89i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.22 + 6.95i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 - 6.80iT - 53T^{2} \) |
| 59 | \( 1 + (-8.57 + 3.11i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.832 - 4.71i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (7.06 - 8.42i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-3.98 + 6.89i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.92 - 3.33i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.28 + 7.49i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (1.28 - 1.08i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (10.9 - 6.30i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.74 - 1.36i)T + (74.3 + 62.3i)T^{2} \) |
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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
−13.44560575109334647667649340978, −12.48736026445499480865834635293, −11.61425276031205907556746618964, −9.773495851246958720796209469728, −9.112615037994135591196775501901, −8.231478932897034126552080081633, −7.36569185329061612675749533122, −5.76063709431973621572363829727, −3.69416925941625129206856975511, −1.40818117106805115600113524779,
2.78885851324175433319746170342, 3.58806274903295636680859668963, 6.49848146216492270006628465479, 7.48036038738794639740321008481, 8.406695202492972785423401164992, 9.803952700903250268878692914813, 10.40726092543171173389255122766, 11.37017997406079411589385396330, 12.70616922722646836361316796022, 14.19820958626008072090099709815
