| L(s) = 1 | + (−0.00960 + 1.41i)2-s + (−0.740 + 0.671i)3-s + (−1.99 − 0.0271i)4-s + (0.0344 + 0.137i)5-s + (−0.942 − 1.05i)6-s + (−1.05 − 0.865i)7-s + (0.0576 − 2.82i)8-s + (0.0980 − 0.995i)9-s + (−0.194 + 0.0473i)10-s + (2.22 − 1.05i)11-s + (1.50 − 1.32i)12-s + (4.08 − 2.45i)13-s + (1.23 − 1.48i)14-s + (−0.117 − 0.0787i)15-s + (3.99 + 0.108i)16-s + (−0.802 + 0.536i)17-s + ⋯ |
| L(s) = 1 | + (−0.00678 + 0.999i)2-s + (−0.427 + 0.387i)3-s + (−0.999 − 0.0135i)4-s + (0.0154 + 0.0614i)5-s + (−0.384 − 0.430i)6-s + (−0.398 − 0.327i)7-s + (0.0203 − 0.999i)8-s + (0.0326 − 0.331i)9-s + (−0.0615 + 0.0149i)10-s + (0.670 − 0.317i)11-s + (0.433 − 0.381i)12-s + (1.13 − 0.679i)13-s + (0.329 − 0.396i)14-s + (−0.0304 − 0.0203i)15-s + (0.999 + 0.0271i)16-s + (−0.194 + 0.130i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04145 + 0.536892i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04145 + 0.536892i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.00960 - 1.41i)T \) |
| 3 | \( 1 + (0.740 - 0.671i)T \) |
| good | 5 | \( 1 + (-0.0344 - 0.137i)T + (-4.40 + 2.35i)T^{2} \) |
| 7 | \( 1 + (1.05 + 0.865i)T + (1.36 + 6.86i)T^{2} \) |
| 11 | \( 1 + (-2.22 + 1.05i)T + (6.97 - 8.50i)T^{2} \) |
| 13 | \( 1 + (-4.08 + 2.45i)T + (6.12 - 11.4i)T^{2} \) |
| 17 | \( 1 + (0.802 - 0.536i)T + (6.50 - 15.7i)T^{2} \) |
| 19 | \( 1 + (-0.299 + 0.404i)T + (-5.51 - 18.1i)T^{2} \) |
| 23 | \( 1 + (1.27 - 0.385i)T + (19.1 - 12.7i)T^{2} \) |
| 29 | \( 1 + (2.31 - 0.828i)T + (22.4 - 18.3i)T^{2} \) |
| 31 | \( 1 + (-1.69 - 4.08i)T + (-21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (-0.599 + 4.04i)T + (-35.4 - 10.7i)T^{2} \) |
| 41 | \( 1 + (-10.1 - 5.41i)T + (22.7 + 34.0i)T^{2} \) |
| 43 | \( 1 + (2.07 - 2.29i)T + (-4.21 - 42.7i)T^{2} \) |
| 47 | \( 1 + (-0.932 + 4.68i)T + (-43.4 - 17.9i)T^{2} \) |
| 53 | \( 1 + (-7.90 - 2.82i)T + (40.9 + 33.6i)T^{2} \) |
| 59 | \( 1 + (2.96 + 1.77i)T + (27.8 + 52.0i)T^{2} \) |
| 61 | \( 1 + (3.51 + 0.172i)T + (60.7 + 5.97i)T^{2} \) |
| 67 | \( 1 + (-0.197 + 4.01i)T + (-66.6 - 6.56i)T^{2} \) |
| 71 | \( 1 + (-5.26 + 0.518i)T + (69.6 - 13.8i)T^{2} \) |
| 73 | \( 1 + (-8.18 + 6.72i)T + (14.2 - 71.5i)T^{2} \) |
| 79 | \( 1 + (0.823 - 0.163i)T + (72.9 - 30.2i)T^{2} \) |
| 83 | \( 1 + (-1.77 - 11.9i)T + (-79.4 + 24.0i)T^{2} \) |
| 89 | \( 1 + (-14.6 - 4.43i)T + (74.0 + 49.4i)T^{2} \) |
| 97 | \( 1 + (-12.7 + 5.27i)T + (68.5 - 68.5i)T^{2} \) |
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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
−10.40708122265421995524024358482, −9.424158714701448392879970489112, −8.734292422246440844186756834929, −7.84819853745366575521874313900, −6.71077790428625383605417589702, −6.18526265140518345523138500675, −5.26974369220359542430456677816, −4.15855418511246679489817680053, −3.36284917952693498732689042502, −0.849674879075178835873801443941,
1.09584648366458624726814036928, 2.30317955054251067794231936245, 3.63940092257117558087045632259, 4.53871540190098730688815471680, 5.74231785763185777240745102114, 6.52948199579379889297717619054, 7.71560595254997337950208871274, 8.868056118933572271822457169286, 9.301231264455047982447679277357, 10.35218331829357360870301910276
