| L(s) = 1 | + (−1.16 − 0.796i)2-s + (1.29 + 1.15i)3-s + (0.732 + 1.86i)4-s + (−0.605 + 2.06i)5-s + (−0.590 − 2.37i)6-s + (−0.738 − 0.639i)7-s + (0.626 − 2.75i)8-s + (0.336 + 2.98i)9-s + (2.35 − 1.92i)10-s + (−0.357 + 0.229i)11-s + (−1.20 + 3.24i)12-s + (2.49 + 2.87i)13-s + (0.353 + 1.33i)14-s + (−3.16 + 1.96i)15-s + (−2.92 + 2.72i)16-s + (−2.23 − 1.02i)17-s + ⋯ |
| L(s) = 1 | + (−0.826 − 0.562i)2-s + (0.745 + 0.666i)3-s + (0.366 + 0.930i)4-s + (−0.270 + 0.922i)5-s + (−0.241 − 0.970i)6-s + (−0.278 − 0.241i)7-s + (0.221 − 0.975i)8-s + (0.112 + 0.993i)9-s + (0.743 − 0.610i)10-s + (−0.107 + 0.0692i)11-s + (−0.347 + 0.937i)12-s + (0.691 + 0.798i)13-s + (0.0944 + 0.356i)14-s + (−0.816 + 0.507i)15-s + (−0.731 + 0.681i)16-s + (−0.542 − 0.247i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.303 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.780723 + 0.570727i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.780723 + 0.570727i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.16 + 0.796i)T \) |
| 3 | \( 1 + (-1.29 - 1.15i)T \) |
| 23 | \( 1 + (-4.60 - 1.32i)T \) |
| good | 5 | \( 1 + (0.605 - 2.06i)T + (-4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (0.738 + 0.639i)T + (0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (0.357 - 0.229i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.49 - 2.87i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (2.23 + 1.02i)T + (11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (6.48 - 2.96i)T + (12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (0.940 + 0.429i)T + (18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-7.16 - 1.02i)T + (29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-3.69 + 1.08i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-3.07 + 10.4i)T + (-34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (5.31 - 0.764i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 3.75T + 47T^{2} \) |
| 53 | \( 1 + (-6.52 - 5.65i)T + (7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-2.92 - 3.37i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.830 + 5.77i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-2.37 + 3.69i)T + (-27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (8.76 + 5.63i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (1.76 + 3.86i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-5.26 + 4.56i)T + (11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (6.94 - 2.03i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-14.0 + 2.02i)T + (85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-6.11 - 1.79i)T + (81.6 + 52.4i)T^{2} \) |
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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
−11.68820405758785973099531570958, −10.75642232729278225994694512942, −10.36784880555640925987705754562, −9.196102725228796743046874611641, −8.533377933126961731199977174728, −7.41006060729077957646717568178, −6.51917654225619133409883560771, −4.30669528659871328288603201058, −3.39800482253342334365386202538, −2.22611013341067937359329198916,
0.921342020060836133932141478939, 2.62723813405890703191757703235, 4.55346954427414385832063943332, 6.04934603556625310257474274220, 6.88596555220879967703141756920, 8.247527464474148437938446928122, 8.496641896871821102296303795601, 9.372930682300772687146329556790, 10.56576901357653771464702737740, 11.66526860037374808437593281132
