| L(s) = 1 | + (0.415 − 0.909i)2-s + (−0.195 − 0.0575i)3-s + (−0.654 − 0.755i)4-s + (−1.56 + 1.00i)5-s + (−0.133 + 0.154i)6-s + (0.142 − 0.989i)7-s + (−0.959 + 0.281i)8-s + (−2.48 − 1.59i)9-s + (0.265 + 1.84i)10-s + (−2.33 − 5.11i)11-s + (0.0848 + 0.185i)12-s + (−0.665 − 4.62i)13-s + (−0.841 − 0.540i)14-s + (0.364 − 0.107i)15-s + (−0.142 + 0.989i)16-s + (2.86 − 3.30i)17-s + ⋯ |
| L(s) = 1 | + (0.293 − 0.643i)2-s + (−0.113 − 0.0332i)3-s + (−0.327 − 0.377i)4-s + (−0.700 + 0.450i)5-s + (−0.0545 + 0.0629i)6-s + (0.0537 − 0.374i)7-s + (−0.339 + 0.0996i)8-s + (−0.829 − 0.533i)9-s + (0.0838 + 0.582i)10-s + (−0.703 − 1.54i)11-s + (0.0244 + 0.0536i)12-s + (−0.184 − 1.28i)13-s + (−0.224 − 0.144i)14-s + (0.0941 − 0.0276i)15-s + (−0.0355 + 0.247i)16-s + (0.695 − 0.802i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.858 + 0.512i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.858 + 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.230594 - 0.836545i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.230594 - 0.836545i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.415 + 0.909i)T \) |
| 7 | \( 1 + (-0.142 + 0.989i)T \) |
| 23 | \( 1 + (4.17 - 2.35i)T \) |
| good | 3 | \( 1 + (0.195 + 0.0575i)T + (2.52 + 1.62i)T^{2} \) |
| 5 | \( 1 + (1.56 - 1.00i)T + (2.07 - 4.54i)T^{2} \) |
| 11 | \( 1 + (2.33 + 5.11i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (0.665 + 4.62i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-2.86 + 3.30i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-3.79 - 4.37i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (0.269 - 0.311i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-0.252 + 0.0741i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-7.85 - 5.05i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (-4.44 + 2.85i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (6.06 + 1.77i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 - 0.945T + 47T^{2} \) |
| 53 | \( 1 + (-0.751 + 5.22i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (1.34 + 9.38i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (4.55 - 1.33i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-5.30 + 11.6i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (3.10 - 6.80i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-7.28 - 8.40i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (1.93 + 13.4i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (9.76 + 6.27i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-4.75 - 1.39i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-14.1 + 9.11i)T + (40.2 - 88.2i)T^{2} \) |
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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
−11.41709740666879009328259113670, −10.52323723840281901688384727181, −9.647057903370879754501290918174, −8.223762440865518515053659579617, −7.67321554104047503998850802066, −5.99527562588540511282377343815, −5.31450793667514637430551828366, −3.47777943494357398106885198996, −3.10478317924594206825259204531, −0.56502559059599166701912485776,
2.42957670686098912782228233981, 4.23718168879569642935662094604, 4.94524324948006678320505923913, 6.07911580568653930137097076258, 7.37406981867544818823006879641, 8.013410643051995043501104240201, 9.025954148250233941118394677712, 10.01356857467679414690725292272, 11.37261024064473688750032061143, 12.09295731783080235569603594665
