| L(s) = 1 | + (−0.849 + 1.13i)2-s + (0.209 + 1.39i)3-s + (−0.556 − 1.92i)4-s + (−1.73 − 0.679i)5-s + (−1.75 − 0.945i)6-s + (−0.592 − 2.57i)7-s + (2.64 + 1.00i)8-s + (0.971 − 0.299i)9-s + (2.23 − 1.38i)10-s + (0.224 − 0.728i)11-s + (2.55 − 1.17i)12-s + (−1.74 + 0.398i)13-s + (3.41 + 1.52i)14-s + (0.582 − 2.55i)15-s + (−3.38 + 2.13i)16-s + (4.21 − 2.87i)17-s + ⋯ |
| L(s) = 1 | + (−0.600 + 0.799i)2-s + (0.121 + 0.803i)3-s + (−0.278 − 0.960i)4-s + (−0.774 − 0.303i)5-s + (−0.715 − 0.386i)6-s + (−0.223 − 0.974i)7-s + (0.935 + 0.354i)8-s + (0.323 − 0.0999i)9-s + (0.708 − 0.436i)10-s + (0.0677 − 0.219i)11-s + (0.738 − 0.340i)12-s + (−0.483 + 0.110i)13-s + (0.913 + 0.406i)14-s + (0.150 − 0.659i)15-s + (−0.845 + 0.534i)16-s + (1.02 − 0.697i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.131i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.861735 + 0.0569109i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.861735 + 0.0569109i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.849 - 1.13i)T \) |
| 7 | \( 1 + (0.592 + 2.57i)T \) |
| good | 3 | \( 1 + (-0.209 - 1.39i)T + (-2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (1.73 + 0.679i)T + (3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (-0.224 + 0.728i)T + (-9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (1.74 - 0.398i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-4.21 + 2.87i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (-7.23 + 4.17i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.115 + 0.0785i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (2.76 - 5.74i)T + (-18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (-3.93 + 6.81i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.03 - 0.302i)T + (36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (-2.31 - 2.90i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (6.71 + 5.35i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (-4.15 + 3.85i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (-7.02 - 0.526i)T + (52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (-7.74 + 3.03i)T + (43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (-7.47 + 0.559i)T + (60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (6.85 + 3.95i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.31 + 1.11i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-0.0710 - 0.0659i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (2.79 + 4.84i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (9.51 + 2.17i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (0.741 - 0.228i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−11.09399359277859845084894184567, −9.958363108270940147429071462102, −9.681861923956831561194094862668, −8.569728816183892082060015689330, −7.41873411996588737933705718105, −7.07635654347395981640916378346, −5.39193215633354922121870076915, −4.53995044582962569321774992295, −3.49069251827243451811721691804, −0.789267122976103784668601081304,
1.45407424742609225828618572700, 2.80806327267107922790785642515, 3.87836637587155840858703299742, 5.48907671634483082484146355720, 7.00456538822969180224004311706, 7.74907366380412677186142903923, 8.361556780733834692949738704033, 9.662114680885471814318279339929, 10.22412545386091933719353566111, 11.62836012340190836366856073298
