| L(s) = 1 | + (−0.439 − 0.487i)2-s + (1.74 + 0.777i)3-s + (0.163 − 1.56i)4-s + (3.49 + 0.742i)5-s + (−0.388 − 1.19i)6-s + (0.295 + 2.62i)7-s + (−1.89 + 1.37i)8-s + (0.439 + 0.487i)9-s + (−1.17 − 2.02i)10-s + (1.49 − 2.59i)12-s + (1.82 − 5.62i)13-s + (1.15 − 1.29i)14-s + (5.52 + 4.01i)15-s + (−1.56 − 0.332i)16-s + (1.10 − 1.23i)17-s + (0.0450 − 0.428i)18-s + ⋯ |
| L(s) = 1 | + (−0.310 − 0.345i)2-s + (1.00 + 0.449i)3-s + (0.0819 − 0.780i)4-s + (1.56 + 0.331i)5-s + (−0.158 − 0.487i)6-s + (0.111 + 0.993i)7-s + (−0.670 + 0.486i)8-s + (0.146 + 0.162i)9-s + (−0.370 − 0.641i)10-s + (0.433 − 0.749i)12-s + (0.506 − 1.55i)13-s + (0.308 − 0.347i)14-s + (1.42 + 1.03i)15-s + (−0.391 − 0.0831i)16-s + (0.268 − 0.298i)17-s + (0.0106 − 0.101i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.38312 - 0.377225i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.38312 - 0.377225i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-0.295 - 2.62i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (0.439 + 0.487i)T + (-0.209 + 1.98i)T^{2} \) |
| 3 | \( 1 + (-1.74 - 0.777i)T + (2.00 + 2.22i)T^{2} \) |
| 5 | \( 1 + (-3.49 - 0.742i)T + (4.56 + 2.03i)T^{2} \) |
| 13 | \( 1 + (-1.82 + 5.62i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-1.10 + 1.23i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.154 - 1.47i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (1.67 - 2.89i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.49 - 1.81i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-6.92 + 1.47i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (4.12 - 1.83i)T + (24.7 - 27.4i)T^{2} \) |
| 41 | \( 1 + (1.04 - 0.755i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 1.59T + 43T^{2} \) |
| 47 | \( 1 + (-0.173 - 1.64i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (9.02 - 1.91i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (0.925 - 8.80i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + (6.54 + 1.39i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-4.91 - 8.50i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.66 + 8.19i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.476 + 4.53i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (-4.27 - 4.75i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (0.0518 + 0.159i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-1.28 + 2.22i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.00 + 9.26i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.04474556021000577343703607867, −9.438864959611315818407253236265, −8.776003216078124106118185864728, −8.003756248075843130507614275214, −6.31200550104124587451482759006, −5.82187392848277954596288265481, −5.05273484418857916994730259121, −3.12071404182273828078995732458, −2.60277131074635544248841599983, −1.48886708448629739769454201027,
1.54696777179572123190286912041, 2.49429506539411098581893481007, 3.68774509514063417172877033060, 4.80627175299332150947155669226, 6.36917971250954118740468205717, 6.76149206406476663950311943794, 7.84768026895258922460655476280, 8.534897970533192990368402254737, 9.195734964020088963936676664004, 9.880225853702446988515618127786
