| L(s) = 1 | + (−1.37 + 0.348i)2-s + (1.48 + 1.60i)3-s + (1.75 − 0.954i)4-s + (−1.25 + 4.05i)5-s + (−2.60 − 1.68i)6-s + (2.55 − 0.685i)7-s + (−2.07 + 1.92i)8-s + (−0.134 + 1.79i)9-s + (0.302 − 5.99i)10-s + (5.31 − 0.398i)11-s + (4.15 + 1.39i)12-s + (−1.33 + 2.77i)13-s + (−3.26 + 1.82i)14-s + (−8.36 + 4.03i)15-s + (2.17 − 3.35i)16-s + (0.221 + 0.0333i)17-s + ⋯ |
| L(s) = 1 | + (−0.969 + 0.246i)2-s + (0.859 + 0.926i)3-s + (0.878 − 0.477i)4-s + (−0.559 + 1.81i)5-s + (−1.06 − 0.686i)6-s + (0.965 − 0.259i)7-s + (−0.734 + 0.679i)8-s + (−0.0447 + 0.596i)9-s + (0.0955 − 1.89i)10-s + (1.60 − 0.120i)11-s + (1.19 + 0.403i)12-s + (−0.370 + 0.768i)13-s + (−0.872 + 0.489i)14-s + (−2.16 + 1.04i)15-s + (0.544 − 0.838i)16-s + (0.0536 + 0.00808i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.490 - 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.490 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.616642 + 1.05402i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.616642 + 1.05402i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.37 - 0.348i)T \) |
| 7 | \( 1 + (-2.55 + 0.685i)T \) |
| good | 3 | \( 1 + (-1.48 - 1.60i)T + (-0.224 + 2.99i)T^{2} \) |
| 5 | \( 1 + (1.25 - 4.05i)T + (-4.13 - 2.81i)T^{2} \) |
| 11 | \( 1 + (-5.31 + 0.398i)T + (10.8 - 1.63i)T^{2} \) |
| 13 | \( 1 + (1.33 - 2.77i)T + (-8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (-0.221 - 0.0333i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (4.52 + 2.61i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.76 + 0.416i)T + (21.9 - 6.77i)T^{2} \) |
| 29 | \( 1 + (1.75 + 1.40i)T + (6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (0.285 + 0.495i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.64 + 1.03i)T + (27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (0.686 + 3.00i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (7.43 + 1.69i)T + (38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (9.37 - 6.39i)T + (17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (0.164 + 0.0644i)T + (38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-2.49 - 8.07i)T + (-48.7 + 33.2i)T^{2} \) |
| 61 | \( 1 + (1.03 - 0.405i)T + (44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (-11.4 + 6.60i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.725 + 0.909i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-6.74 - 4.59i)T + (26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + (-4.67 + 8.10i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.51 - 3.14i)T + (-51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (-0.794 + 10.6i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 - 8.28T + 97T^{2} \) |
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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
−11.27910294748394368046863614295, −10.61141734155437026906586024302, −9.712060746997263496735369665943, −8.924552617847397827309489185522, −8.038354294700701552302893329529, −7.00334253884300930881280111346, −6.44924866148116326571777418037, −4.39417763137186847285516861577, −3.41191187708020679892903506027, −2.16177609289996110270591781392,
1.13157618594780005197686879737, 1.90574707472447216804319077839, 3.73358313082184663331004784882, 5.06493992353876374695359894833, 6.66491638271059025109997338164, 7.85116381640970028235407289444, 8.287972449951570549320919659286, 8.832341583970183871992378700825, 9.670054769439745537002750615921, 11.22217128505240353441884592779
