L(s) = 1 | + (1.27 − 0.601i)2-s + (0.337 − 0.139i)3-s + (1.27 − 1.54i)4-s + (1.59 − 1.59i)5-s + (0.348 − 0.382i)6-s + (−3.95 + 1.63i)7-s + (0.704 − 2.73i)8-s + (−2.02 + 2.02i)9-s + (1.08 − 3.00i)10-s + (0.482 + 1.16i)11-s + (0.215 − 0.699i)12-s + (1.71 + 4.13i)13-s + (−4.07 + 4.47i)14-s + (0.316 − 0.763i)15-s + (−0.746 − 3.92i)16-s + (1.72 − 4.16i)17-s + ⋯ |
L(s) = 1 | + (0.904 − 0.425i)2-s + (0.195 − 0.0808i)3-s + (0.637 − 0.770i)4-s + (0.714 − 0.714i)5-s + (0.142 − 0.156i)6-s + (−1.49 + 0.619i)7-s + (0.249 − 0.968i)8-s + (−0.675 + 0.675i)9-s + (0.342 − 0.951i)10-s + (0.145 + 0.350i)11-s + (0.0621 − 0.201i)12-s + (0.475 + 1.14i)13-s + (−1.08 + 1.19i)14-s + (0.0817 − 0.197i)15-s + (−0.186 − 0.982i)16-s + (0.418 − 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.663 + 0.747i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.663 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.71010 - 0.768513i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.71010 - 0.768513i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.27 + 0.601i)T \) |
| 41 | \( 1 + (6.23 + 1.44i)T \) |
good | 3 | \( 1 + (-0.337 + 0.139i)T + (2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (-1.59 + 1.59i)T - 5iT^{2} \) |
| 7 | \( 1 + (3.95 - 1.63i)T + (4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.482 - 1.16i)T + (-7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (-1.71 - 4.13i)T + (-9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + (-1.72 + 4.16i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-1.71 - 0.708i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 - 0.562T + 23T^{2} \) |
| 29 | \( 1 + (0.0734 + 0.177i)T + (-20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + 8.53T + 31T^{2} \) |
| 37 | \( 1 + 3.69T + 37T^{2} \) |
| 43 | \( 1 + (-8.35 + 8.35i)T - 43iT^{2} \) |
| 47 | \( 1 + (-8.03 - 3.32i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (6.77 - 2.80i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + 6.62iT - 59T^{2} \) |
| 61 | \( 1 + (9.13 + 9.13i)T + 61iT^{2} \) |
| 67 | \( 1 + (3.41 + 1.41i)T + (47.3 + 47.3i)T^{2} \) |
| 71 | \( 1 + (-4.52 + 1.87i)T + (50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (-7.76 - 7.76i)T + 73iT^{2} \) |
| 79 | \( 1 + (2.49 + 6.02i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 - 6.02iT - 83T^{2} \) |
| 89 | \( 1 + (-6.74 - 16.2i)T + (-62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + (-2.18 - 0.906i)T + (68.5 + 68.5i)T^{2} \) |
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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
−12.71967732409163738851906271846, −12.06619257486921698057394805085, −10.87941242495409875833970167374, −9.484723269133206322363558194247, −9.149284847256781198783350245956, −7.11356896953912140698268948225, −5.95011730781968020147114520115, −5.13253085782497697066126883176, −3.44039784607070166044051710020, −2.08543301223258676037542313348,
3.01677454697150314554122078308, 3.62590701909031615818319964261, 5.82926141805494769606318864909, 6.25345413991216820902143235842, 7.42031591829673278132876736084, 8.847295814711228687943349581945, 10.12917874571092316311190084985, 10.94125947521757151653603286536, 12.38139410301332482179433536141, 13.13828252671777231360887119857