| L(s) = 1 | + (−0.173 + 0.984i)2-s + (−0.983 − 0.825i)3-s + (−0.939 − 0.342i)4-s + (−3.47 + 1.26i)5-s + (0.983 − 0.825i)6-s + (0.221 + 0.383i)7-s + (0.5 − 0.866i)8-s + (−0.234 − 1.33i)9-s + (−0.641 − 3.63i)10-s + (2.01 − 3.48i)11-s + (0.642 + 1.11i)12-s + (−3.74 + 3.14i)13-s + (−0.415 + 0.151i)14-s + (4.45 + 1.62i)15-s + (0.766 + 0.642i)16-s + (0.0463 − 0.262i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 + 0.696i)2-s + (−0.567 − 0.476i)3-s + (−0.469 − 0.171i)4-s + (−1.55 + 0.565i)5-s + (0.401 − 0.336i)6-s + (0.0836 + 0.144i)7-s + (0.176 − 0.306i)8-s + (−0.0782 − 0.443i)9-s + (−0.202 − 1.15i)10-s + (0.607 − 1.05i)11-s + (0.185 + 0.321i)12-s + (−1.03 + 0.872i)13-s + (−0.111 + 0.0404i)14-s + (1.15 + 0.419i)15-s + (0.191 + 0.160i)16-s + (0.0112 − 0.0637i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.658844 + 0.253337i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.658844 + 0.253337i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + (0.983 + 0.825i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (3.47 - 1.26i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.221 - 0.383i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.01 + 3.48i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.74 - 3.14i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.0463 + 0.262i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-8.65 - 3.14i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.0388 - 0.220i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.73 - 3.01i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 1.44T + 37T^{2} \) |
| 41 | \( 1 + (-6.02 - 5.05i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-5.29 + 1.92i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.381 - 2.16i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-9.34 - 3.40i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.609 + 3.45i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (3.83 + 1.39i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (0.0256 + 0.145i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-10.7 + 3.92i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-1.08 - 0.914i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (7.81 + 6.55i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.64 - 2.84i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.56 + 3.83i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (0.709 - 4.02i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.78608971223237614177015664897, −9.330658304301766330257575175135, −8.709438228104707495258242968967, −7.59849438067100954116348348050, −7.03485949203491697247809750665, −6.39672180657057106110741142381, −5.22532029208532947570209145807, −4.10096296822792312165562108023, −3.12822718074222839634983357069, −0.78301543510221053331756838733,
0.68774084187042272665822373739, 2.62305836650509624349763296525, 4.03563429059491812464743203446, 4.60216083241490852745141428945, 5.34651402863220760134429973173, 7.13779073399262879935053511718, 7.73229349821693635000045554303, 8.681505307475076025187878121797, 9.592867063323107910400976384925, 10.53564067770628592811452823370
