| L(s) = 1 | + (0.826 − 0.563i)2-s + (−0.733 + 0.680i)3-s + (0.365 − 0.930i)4-s + (−3.12 − 0.963i)5-s + (−0.222 + 0.974i)6-s + (−2.56 + 0.633i)7-s + (−0.222 − 0.974i)8-s + (0.0747 − 0.997i)9-s + (−3.12 + 0.963i)10-s + (−0.402 − 5.36i)11-s + (0.365 + 0.930i)12-s + (−4.97 − 2.39i)13-s + (−1.76 + 1.97i)14-s + (2.94 − 1.41i)15-s + (−0.733 − 0.680i)16-s + (7.74 + 1.16i)17-s + ⋯ |
| L(s) = 1 | + (0.584 − 0.398i)2-s + (−0.423 + 0.392i)3-s + (0.182 − 0.465i)4-s + (−1.39 − 0.430i)5-s + (−0.0908 + 0.398i)6-s + (−0.970 + 0.239i)7-s + (−0.0786 − 0.344i)8-s + (0.0249 − 0.332i)9-s + (−0.987 + 0.304i)10-s + (−0.121 − 1.61i)11-s + (0.105 + 0.268i)12-s + (−1.37 − 0.664i)13-s + (−0.471 + 0.526i)14-s + (0.760 − 0.366i)15-s + (−0.183 − 0.170i)16-s + (1.87 + 0.283i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.106462 - 0.527943i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.106462 - 0.527943i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.826 + 0.563i)T \) |
| 3 | \( 1 + (0.733 - 0.680i)T \) |
| 7 | \( 1 + (2.56 - 0.633i)T \) |
| good | 5 | \( 1 + (3.12 + 0.963i)T + (4.13 + 2.81i)T^{2} \) |
| 11 | \( 1 + (0.402 + 5.36i)T + (-10.8 + 1.63i)T^{2} \) |
| 13 | \( 1 + (4.97 + 2.39i)T + (8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (-7.74 - 1.16i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (2.78 - 4.82i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.63 - 0.246i)T + (21.9 - 6.77i)T^{2} \) |
| 29 | \( 1 + (1.04 - 1.31i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (0.881 + 1.52i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.104 + 0.266i)T + (-27.1 + 25.1i)T^{2} \) |
| 41 | \( 1 + (1.60 + 7.03i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (0.367 - 1.60i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (-0.818 + 0.558i)T + (17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (-2.62 + 6.69i)T + (-38.8 - 36.0i)T^{2} \) |
| 59 | \( 1 + (5.68 - 1.75i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (4.56 + 11.6i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (-3.25 - 5.63i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.76 + 2.20i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-5.78 - 3.94i)T + (26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + (-2.69 + 4.67i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.29 - 0.621i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-0.926 + 12.3i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 + 18.6T + 97T^{2} \) |
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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
−11.59915712422396375869634912480, −10.53584432853308325071412926057, −9.798370307946815012077844650715, −8.441807115816057148349894763702, −7.54441111482159027983573869721, −6.01254618252132829161166903542, −5.26940753465765983856407201845, −3.80929094719842793550274305139, −3.22356943063965021910979414356, −0.33506825582683687680019681422,
2.73373297694651231577313308776, 4.08421986093948394061371013058, 4.99803622182281569190348762905, 6.57127857794890035893026577188, 7.31422475492252728171212484500, 7.69894227092888099359330438045, 9.466591980925497420558221923696, 10.39425533868626753651953919544, 11.70659419192941677192747265345, 12.24526263359352566616203484950
