| L(s) = 1 | + (6.38 − 4.64i)2-s + (3.63 + 11.1i)3-s + (9.38 − 28.8i)4-s + (−51.9 − 37.7i)5-s + (75.1 + 54.6i)6-s + (−33.5 + 103. i)7-s + (3.96 + 12.2i)8-s + (84.5 − 61.4i)9-s − 507.·10-s + (−398. + 48.9i)11-s + 357.·12-s + (830. − 603. i)13-s + (264. + 815. i)14-s + (233. − 719. i)15-s + (868. + 630. i)16-s + (−1.04e3 − 761. i)17-s + ⋯ |
| L(s) = 1 | + (1.12 − 0.820i)2-s + (0.233 + 0.717i)3-s + (0.293 − 0.902i)4-s + (−0.930 − 0.675i)5-s + (0.852 + 0.619i)6-s + (−0.258 + 0.796i)7-s + (0.0219 + 0.0674i)8-s + (0.348 − 0.252i)9-s − 1.60·10-s + (−0.992 + 0.121i)11-s + 0.716·12-s + (1.36 − 0.989i)13-s + (0.361 + 1.11i)14-s + (0.268 − 0.825i)15-s + (0.848 + 0.616i)16-s + (−0.879 − 0.638i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 + 0.505i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.862 + 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.80641 - 0.490075i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.80641 - 0.490075i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (398. - 48.9i)T \) |
| good | 2 | \( 1 + (-6.38 + 4.64i)T + (9.88 - 30.4i)T^{2} \) |
| 3 | \( 1 + (-3.63 - 11.1i)T + (-196. + 142. i)T^{2} \) |
| 5 | \( 1 + (51.9 + 37.7i)T + (965. + 2.97e3i)T^{2} \) |
| 7 | \( 1 + (33.5 - 103. i)T + (-1.35e4 - 9.87e3i)T^{2} \) |
| 13 | \( 1 + (-830. + 603. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (1.04e3 + 761. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-105. - 323. i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 - 817.T + 6.43e6T^{2} \) |
| 29 | \( 1 + (1.22e3 - 3.76e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (3.17e3 - 2.30e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (3.79e3 - 1.16e4i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (4.31e3 + 1.32e4i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 + 691.T + 1.47e8T^{2} \) |
| 47 | \( 1 + (4.53e3 + 1.39e4i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (1.15e3 - 841. i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-2.26e3 + 6.95e3i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (1.40e4 + 1.02e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + 1.69e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-5.74e4 - 4.17e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-1.09e4 + 3.36e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-5.43e4 + 3.94e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (4.41e4 + 3.20e4i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 - 5.78e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-6.15e4 + 4.47e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−20.22637995055875535438911415026, −18.40745447902408227075597233722, −15.89066227945871580141123882683, −15.29649671027389468544813244694, −13.24831472610680702281731935601, −12.23971148625891191345662880704, −10.73070235256408395708157930738, −8.587080606969702581762244227690, −5.06005600372756231296049622787, −3.43912514593256884474712428362,
3.99537000271162612461581914533, 6.62091473661192115210851142758, 7.71773840940771996387363300930, 10.95619795824376146662424130951, 13.01832088979434467238170881015, 13.77141435762420603158715808790, 15.32393077381371075175316002698, 16.28224587185045269160965456780, 18.49161907429363130906643226878, 19.46758948553030187009466433450
