| L(s) = 1 | + (0.191 + 1.82i)2-s + (−1.47 + 1.63i)3-s + (−1.33 + 0.283i)4-s + (−0.580 + 0.258i)5-s + (−3.26 − 2.37i)6-s + (2.39 + 1.11i)7-s + (0.360 + 1.10i)8-s + (−0.191 − 1.82i)9-s + (−0.582 − 1.00i)10-s + (1.5 − 2.59i)12-s + (−1.45 + 1.05i)13-s + (−1.57 + 4.59i)14-s + (0.431 − 1.32i)15-s + (−4.44 + 1.97i)16-s + (−0.296 + 2.81i)17-s + (3.29 − 0.699i)18-s + ⋯ |
| L(s) = 1 | + (0.135 + 1.28i)2-s + (−0.849 + 0.943i)3-s + (−0.667 + 0.141i)4-s + (−0.259 + 0.115i)5-s + (−1.33 − 0.967i)6-s + (0.907 + 0.421i)7-s + (0.127 + 0.391i)8-s + (−0.0639 − 0.608i)9-s + (−0.184 − 0.319i)10-s + (0.433 − 0.749i)12-s + (−0.404 + 0.293i)13-s + (−0.420 + 1.22i)14-s + (0.111 − 0.343i)15-s + (−1.11 + 0.494i)16-s + (−0.0718 + 0.683i)17-s + (0.775 − 0.164i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.321 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.321 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.567911 - 0.792487i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.567911 - 0.792487i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.39 - 1.11i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (-0.191 - 1.82i)T + (-1.95 + 0.415i)T^{2} \) |
| 3 | \( 1 + (1.47 - 1.63i)T + (-0.313 - 2.98i)T^{2} \) |
| 5 | \( 1 + (0.580 - 0.258i)T + (3.34 - 3.71i)T^{2} \) |
| 13 | \( 1 + (1.45 - 1.05i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.296 - 2.81i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (5.44 + 1.15i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (1.08 - 1.87i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.22 + 9.91i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-5.87 - 2.61i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (-4.05 - 4.50i)T + (-3.86 + 36.7i)T^{2} \) |
| 41 | \( 1 + (2.32 + 7.16i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 4.86T + 43T^{2} \) |
| 47 | \( 1 + (-2.77 - 0.589i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-6.82 - 3.03i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (11.5 - 2.45i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-3.95 + 1.76i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (-0.801 - 1.38i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.47 + 2.52i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (15.6 - 3.32i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (-0.497 - 4.73i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (7.46 + 5.42i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (0.182 - 0.315i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.10 + 1.52i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.78573494515355923555615280034, −10.00286836694432510214587553618, −8.838400223601673511897570665224, −8.119854986087729526891912560485, −7.32643851677349390559071962292, −6.14277375949301595936347171538, −5.72058125587844047580369639788, −4.60348599439188198001003139194, −4.28663352352746340415933292173, −2.21240589490226800779003607244,
0.50793947432956790105375344217, 1.57594831552389736102466087283, 2.65664175200191083203277667514, 4.12200054558546764479870030002, 4.86000820281074450070151309076, 6.13570565609211537173008153240, 7.03187752276798466215278717627, 7.79239958025166076475323761823, 8.822957689367216757137375760913, 10.08610293115486113146397593259
