L(s) = 1 | + (1.12 − 0.860i)2-s + (−2.97 − 1.01i)3-s + (0.520 − 1.93i)4-s + (1.52 − 3.08i)5-s + (−4.21 + 1.42i)6-s + (−0.574 − 2.58i)7-s + (−1.07 − 2.61i)8-s + (5.46 + 4.19i)9-s + (−0.945 − 4.76i)10-s + (1.53 + 1.34i)11-s + (−3.50 + 5.22i)12-s + (−1.18 + 1.78i)13-s + (−2.86 − 2.40i)14-s + (−7.64 + 7.64i)15-s + (−3.45 − 2.00i)16-s + (5.29 − 1.41i)17-s + ⋯ |
L(s) = 1 | + (0.793 − 0.608i)2-s + (−1.71 − 0.583i)3-s + (0.260 − 0.965i)4-s + (0.680 − 1.37i)5-s + (−1.71 + 0.582i)6-s + (−0.217 − 0.976i)7-s + (−0.380 − 0.924i)8-s + (1.82 + 1.39i)9-s + (−0.299 − 1.50i)10-s + (0.463 + 0.406i)11-s + (−1.01 + 1.50i)12-s + (−0.329 + 0.493i)13-s + (−0.766 − 0.642i)14-s + (−1.97 + 1.97i)15-s + (−0.864 − 0.502i)16-s + (1.28 − 0.343i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0536296 + 1.28394i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0536296 + 1.28394i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.12 + 0.860i)T \) |
| 7 | \( 1 + (0.574 + 2.58i)T \) |
good | 3 | \( 1 + (2.97 + 1.01i)T + (2.38 + 1.82i)T^{2} \) |
| 5 | \( 1 + (-1.52 + 3.08i)T + (-3.04 - 3.96i)T^{2} \) |
| 11 | \( 1 + (-1.53 - 1.34i)T + (1.43 + 10.9i)T^{2} \) |
| 13 | \( 1 + (1.18 - 1.78i)T + (-4.97 - 12.0i)T^{2} \) |
| 17 | \( 1 + (-5.29 + 1.41i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (6.10 + 0.399i)T + (18.8 + 2.47i)T^{2} \) |
| 23 | \( 1 + (-5.26 - 4.04i)T + (5.95 + 22.2i)T^{2} \) |
| 29 | \( 1 + (4.44 - 0.884i)T + (26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 + (-0.846 - 0.488i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.466 - 0.946i)T + (-22.5 - 29.3i)T^{2} \) |
| 41 | \( 1 + (-2.49 + 1.03i)T + (28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (-10.3 - 2.06i)T + (39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (-0.388 - 0.104i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-3.16 + 3.60i)T + (-6.91 - 52.5i)T^{2} \) |
| 59 | \( 1 + (0.718 + 10.9i)T + (-58.4 + 7.70i)T^{2} \) |
| 61 | \( 1 + (-5.44 - 6.20i)T + (-7.96 + 60.4i)T^{2} \) |
| 67 | \( 1 + (-3.91 + 11.5i)T + (-53.1 - 40.7i)T^{2} \) |
| 71 | \( 1 + (2.42 - 5.85i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (3.89 + 0.512i)T + (70.5 + 18.8i)T^{2} \) |
| 79 | \( 1 + (3.28 + 0.880i)T + (68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (1.87 - 2.80i)T + (-31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (0.208 + 1.58i)T + (-85.9 + 23.0i)T^{2} \) |
| 97 | \( 1 - 5.63T + 97T^{2} \) |
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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
−10.91382281863003167088840700591, −10.01963175987296195489929987354, −9.294205430630443954154722565238, −7.40471499725224706577907172318, −6.56333357634779500531521559675, −5.63411287755999916663418892879, −4.94312706221115242329453188370, −4.13900334140141056954729007103, −1.69515357502618373924172481559, −0.813312027357837218511756271637,
2.67512612664094501883117001821, 3.95176741456272857126423649084, 5.28386057490098265355468390725, 5.97754751569408025979691029104, 6.36147059319053014002753427227, 7.34238992250059571064905642121, 8.948925616759005620340462453363, 10.15462673344784208422165980854, 10.80707671439005974771504986819, 11.53441404277698570681943515024