| L(s) = 1 | + (0.173 − 0.984i)2-s + (−0.766 − 0.642i)3-s + (−0.939 − 0.342i)4-s + (2.20 − 0.802i)5-s + (−0.766 + 0.642i)6-s + (−1.78 − 3.09i)7-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.407 − 2.31i)10-s + (−1.35 + 2.35i)11-s + (0.499 + 0.866i)12-s + (4.14 − 3.47i)13-s + (−3.35 + 1.22i)14-s + (−2.20 − 0.802i)15-s + (0.766 + 0.642i)16-s + (−0.673 + 3.82i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (−0.442 − 0.371i)3-s + (−0.469 − 0.171i)4-s + (0.986 − 0.359i)5-s + (−0.312 + 0.262i)6-s + (−0.675 − 1.16i)7-s + (−0.176 + 0.306i)8-s + (0.0578 + 0.328i)9-s + (−0.128 − 0.731i)10-s + (−0.409 + 0.709i)11-s + (0.144 + 0.249i)12-s + (1.14 − 0.964i)13-s + (−0.897 + 0.326i)14-s + (−0.569 − 0.207i)15-s + (0.191 + 0.160i)16-s + (−0.163 + 0.926i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.667562 - 0.761107i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.667562 - 0.761107i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 3 | \( 1 + (0.766 + 0.642i)T \) |
| 19 | \( 1 + (-1.01 - 4.23i)T \) |
| good | 5 | \( 1 + (-2.20 + 0.802i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (1.78 + 3.09i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.35 - 2.35i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.14 + 3.47i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.673 - 3.82i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-7.73 - 2.81i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.613 - 3.47i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (3.26 + 5.65i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 0.389T + 37T^{2} \) |
| 41 | \( 1 + (1.48 + 1.24i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (4.71 - 1.71i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.518 - 2.94i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (7.80 + 2.84i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.474 + 2.68i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (5.91 + 2.15i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (2.59 + 14.7i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (8.47 - 3.08i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-7.88 - 6.61i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-9.96 - 8.36i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-4.08 - 7.07i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-8.98 + 7.53i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (1.49 - 8.47i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−13.06136804824668497460622927832, −12.70370814224722694508997915358, −11.00035668224576954739634471561, −10.34684086294071786509314286657, −9.409355787100665606930894958816, −7.83805418132432336909456888819, −6.37797401649676155501937940650, −5.25724144962815790430590636902, −3.56332128718559954522635432915, −1.42302436248626201470600860177,
2.96397647136189785474413584675, 5.02949662776324935903552890573, 6.05060700637369234355671370711, 6.77539340786880919906913353075, 8.852023872903113662722385557184, 9.302322995052587453371053310822, 10.70461624818910396087271429552, 11.78634732408212933642925376916, 13.17832052223045549105730606678, 13.78988079276357907587262916400
