| 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
−13.78988079276357907587262916400, −13.17832052223045549105730606678, −11.78634732408212933642925376916, −10.70461624818910396087271429552, −9.302322995052587453371053310822, −8.852023872903113662722385557184, −6.77539340786880919906913353075, −6.05060700637369234355671370711, −5.02949662776324935903552890573, −2.96397647136189785474413584675,
1.42302436248626201470600860177, 3.56332128718559954522635432915, 5.25724144962815790430590636902, 6.37797401649676155501937940650, 7.83805418132432336909456888819, 9.409355787100665606930894958816, 10.34684086294071786509314286657, 11.00035668224576954739634471561, 12.70370814224722694508997915358, 13.06136804824668497460622927832
