| L(s) = 1 | − 20.4i·2-s + (−201. + 116. i)3-s + 607.·4-s + (4.87e3 − 2.81e3i)5-s + (2.37e3 + 4.11e3i)6-s + (−6.54e3 − 3.77e3i)7-s − 3.33e4i·8-s + (−2.41e3 + 4.18e3i)9-s + (−5.74e4 − 9.94e4i)10-s + 3.09e5·11-s + (−1.22e5 + 7.06e4i)12-s + (−1.68e5 + 2.91e5i)13-s + (−7.71e4 + 1.33e5i)14-s + (−6.55e5 + 1.13e6i)15-s − 5.81e4·16-s + (1.17e5 − 2.03e5i)17-s + ⋯ |
| L(s) = 1 | − 0.638i·2-s + (−0.829 + 0.479i)3-s + 0.592·4-s + (1.55 − 0.900i)5-s + (0.305 + 0.529i)6-s + (−0.389 − 0.224i)7-s − 1.01i·8-s + (−0.0409 + 0.0709i)9-s + (−0.574 − 0.994i)10-s + 1.92·11-s + (−0.492 + 0.284i)12-s + (−0.453 + 0.785i)13-s + (−0.143 + 0.248i)14-s + (−0.862 + 1.49i)15-s − 0.0554·16-s + (0.0828 − 0.143i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.347 + 0.937i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(2.02366 - 1.40774i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.02366 - 1.40774i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.41e8 - 3.90e7i)T \) |
| good | 2 | \( 1 + 20.4iT - 1.02e3T^{2} \) |
| 3 | \( 1 + (201. - 116. i)T + (2.95e4 - 5.11e4i)T^{2} \) |
| 5 | \( 1 + (-4.87e3 + 2.81e3i)T + (4.88e6 - 8.45e6i)T^{2} \) |
| 7 | \( 1 + (6.54e3 + 3.77e3i)T + (1.41e8 + 2.44e8i)T^{2} \) |
| 11 | \( 1 - 3.09e5T + 2.59e10T^{2} \) |
| 13 | \( 1 + (1.68e5 - 2.91e5i)T + (-6.89e10 - 1.19e11i)T^{2} \) |
| 17 | \( 1 + (-1.17e5 + 2.03e5i)T + (-1.00e12 - 1.74e12i)T^{2} \) |
| 19 | \( 1 + (7.37e5 - 4.25e5i)T + (3.06e12 - 5.30e12i)T^{2} \) |
| 23 | \( 1 + (-2.63e6 - 4.56e6i)T + (-2.07e13 + 3.58e13i)T^{2} \) |
| 29 | \( 1 + (-5.47e6 - 3.16e6i)T + (2.10e14 + 3.64e14i)T^{2} \) |
| 31 | \( 1 + (1.79e7 + 3.11e7i)T + (-4.09e14 + 7.09e14i)T^{2} \) |
| 37 | \( 1 + (-5.89e7 + 3.40e7i)T + (2.40e15 - 4.16e15i)T^{2} \) |
| 41 | \( 1 - 1.60e8T + 1.34e16T^{2} \) |
| 47 | \( 1 - 1.69e8T + 5.25e16T^{2} \) |
| 53 | \( 1 + (6.07e7 + 1.05e8i)T + (-8.74e16 + 1.51e17i)T^{2} \) |
| 59 | \( 1 - 7.67e8T + 5.11e17T^{2} \) |
| 61 | \( 1 + (-6.41e8 - 3.70e8i)T + (3.56e17 + 6.17e17i)T^{2} \) |
| 67 | \( 1 + (8.44e8 + 1.46e9i)T + (-9.11e17 + 1.57e18i)T^{2} \) |
| 71 | \( 1 + (1.33e9 + 7.68e8i)T + (1.62e18 + 2.81e18i)T^{2} \) |
| 73 | \( 1 + (-5.28e8 - 3.05e8i)T + (2.14e18 + 3.72e18i)T^{2} \) |
| 79 | \( 1 + (2.40e9 - 4.16e9i)T + (-4.73e18 - 8.19e18i)T^{2} \) |
| 83 | \( 1 + (-2.27e8 - 3.94e8i)T + (-7.75e18 + 1.34e19i)T^{2} \) |
| 89 | \( 1 + (-4.12e9 + 2.38e9i)T + (1.55e19 - 2.70e19i)T^{2} \) |
| 97 | \( 1 + 1.08e10T + 7.37e19T^{2} \) |
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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
−13.27363064138348203513934094531, −12.11018320296804725825919023045, −11.19671062097704308748630843705, −9.851616419525933410043809659720, −9.290438137833025714635503263744, −6.66989229185396963443776334659, −5.74835730815551301642466520930, −4.19946447828940141529898452243, −2.08849315889114809632628814450, −0.991843325622322225640625542099,
1.35707204909770370115423006700, 2.80064466394335330289739011712, 5.62878548851557907065279344373, 6.40022614820137370153024639189, 6.92804295837299781621484646209, 9.122943461420791923726181235227, 10.47103360680151808885737176136, 11.58347337915092488254728463766, 12.75210708500355302197648197484, 14.35233613739692438425073241175
