| L(s) = 1 | + (0.392 + 1.35i)2-s + (1.07 + 1.35i)3-s + (−1.69 + 1.06i)4-s + (0.219 − 0.144i)5-s + (−1.42 + 1.99i)6-s + (1.89 + 1.78i)7-s + (−2.11 − 1.88i)8-s + (−0.680 + 2.92i)9-s + (0.281 + 0.241i)10-s + (1.21 + 2.41i)11-s + (−3.26 − 1.14i)12-s + (2.54 − 1.09i)13-s + (−1.68 + 3.27i)14-s + (0.431 + 0.142i)15-s + (1.72 − 3.60i)16-s + (2.50 + 2.98i)17-s + ⋯ |
| L(s) = 1 | + (0.277 + 0.960i)2-s + (0.621 + 0.783i)3-s + (−0.846 + 0.532i)4-s + (0.0980 − 0.0644i)5-s + (−0.579 + 0.814i)6-s + (0.715 + 0.675i)7-s + (−0.746 − 0.665i)8-s + (−0.226 + 0.973i)9-s + (0.0891 + 0.0762i)10-s + (0.365 + 0.727i)11-s + (−0.943 − 0.331i)12-s + (0.705 − 0.304i)13-s + (−0.450 + 0.875i)14-s + (0.111 + 0.0366i)15-s + (0.431 − 0.901i)16-s + (0.606 + 0.723i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.380i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 - 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.392088 + 1.98583i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.392088 + 1.98583i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.392 - 1.35i)T \) |
| 3 | \( 1 + (-1.07 - 1.35i)T \) |
| good | 5 | \( 1 + (-0.219 + 0.144i)T + (1.98 - 4.59i)T^{2} \) |
| 7 | \( 1 + (-1.89 - 1.78i)T + (0.407 + 6.98i)T^{2} \) |
| 11 | \( 1 + (-1.21 - 2.41i)T + (-6.56 + 8.82i)T^{2} \) |
| 13 | \( 1 + (-2.54 + 1.09i)T + (8.92 - 9.45i)T^{2} \) |
| 17 | \( 1 + (-2.50 - 2.98i)T + (-2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (3.00 + 2.52i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (6.25 + 6.62i)T + (-1.33 + 22.9i)T^{2} \) |
| 29 | \( 1 + (0.0984 - 0.0115i)T + (28.2 - 6.68i)T^{2} \) |
| 31 | \( 1 + (0.666 + 2.81i)T + (-27.7 + 13.9i)T^{2} \) |
| 37 | \( 1 + (-7.49 + 1.32i)T + (34.7 - 12.6i)T^{2} \) |
| 41 | \( 1 + (5.60 + 4.17i)T + (11.7 + 39.2i)T^{2} \) |
| 43 | \( 1 + (0.165 - 2.84i)T + (-42.7 - 4.99i)T^{2} \) |
| 47 | \( 1 + (-5.56 - 1.31i)T + (42.0 + 21.0i)T^{2} \) |
| 53 | \( 1 + (-0.163 - 0.283i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.24 - 8.45i)T + (-35.2 - 47.3i)T^{2} \) |
| 61 | \( 1 + (-7.91 + 2.36i)T + (50.9 - 33.5i)T^{2} \) |
| 67 | \( 1 + (-14.7 - 1.72i)T + (65.1 + 15.4i)T^{2} \) |
| 71 | \( 1 + (13.5 + 4.93i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-9.95 + 3.62i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-4.64 + 3.46i)T + (22.6 - 75.6i)T^{2} \) |
| 83 | \( 1 + (-8.44 + 6.28i)T + (23.8 - 79.5i)T^{2} \) |
| 89 | \( 1 + (-2.43 - 6.67i)T + (-68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (3.34 + 2.19i)T + (38.4 + 89.0i)T^{2} \) |
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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
−10.74594171428881778713990777041, −9.815868045493893732021461241642, −8.962785940461331089557911610387, −8.320685472401644846342185071328, −7.67168899121833554054486658447, −6.31486163320619710845716706939, −5.46266120856625301785554740588, −4.49037209838394763710021417233, −3.74342671714226619467566674580, −2.22748232215445732358007062867,
1.03012312879111242058207324048, 2.05102353156818848463218515900, 3.43340287736150456132055137889, 4.14046515046237550382291638668, 5.61356840740284762432103151466, 6.50416123772016958178713931608, 7.87149223601823628570946836030, 8.374504931911989541058572142342, 9.420464111025616811751683446332, 10.20672489698500093110172411359
