| L(s) = 1 | + (−0.984 − 0.173i)2-s + (−0.266 − 1.50i)3-s + (0.939 + 0.342i)4-s + (−1.97 − 2.35i)5-s + 1.53i·6-s + (0.153 − 0.128i)7-s + (−0.866 − 0.5i)8-s + (0.613 − 0.223i)9-s + (1.53 + 2.66i)10-s + (2.17 − 3.76i)11-s + (0.266 − 1.50i)12-s + (−1.59 + 4.38i)13-s + (−0.173 + 0.0999i)14-s + (−3.03 + 3.61i)15-s + (0.766 + 0.642i)16-s + (2.32 + 6.38i)17-s + ⋯ |
| L(s) = 1 | + (−0.696 − 0.122i)2-s + (−0.153 − 0.871i)3-s + (0.469 + 0.171i)4-s + (−0.885 − 1.05i)5-s + 0.625i·6-s + (0.0578 − 0.0485i)7-s + (−0.306 − 0.176i)8-s + (0.204 − 0.0744i)9-s + (0.486 + 0.843i)10-s + (0.654 − 1.13i)11-s + (0.0768 − 0.435i)12-s + (−0.443 + 1.21i)13-s + (−0.0462 + 0.0267i)14-s + (−0.783 + 0.933i)15-s + (0.191 + 0.160i)16-s + (0.563 + 1.54i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0469 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0469 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.424065 - 0.444454i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.424065 - 0.444454i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.984 + 0.173i)T \) |
| 37 | \( 1 + (-5.64 + 2.27i)T \) |
| good | 3 | \( 1 + (0.266 + 1.50i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (1.97 + 2.35i)T + (-0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.153 + 0.128i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-2.17 + 3.76i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.59 - 4.38i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.32 - 6.38i)T + (-13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (-4.07 + 0.719i)T + (17.8 - 6.49i)T^{2} \) |
| 23 | \( 1 + (0.896 - 0.517i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.25 - 0.727i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 5.10iT - 31T^{2} \) |
| 41 | \( 1 + (4.46 + 1.62i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 - 0.399iT - 43T^{2} \) |
| 47 | \( 1 + (-4.10 - 7.10i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (8.65 + 7.26i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (2.69 - 3.21i)T + (-10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (3.60 - 9.89i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-6.67 + 5.59i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.45 - 13.9i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 - 7.27T + 73T^{2} \) |
| 79 | \( 1 + (4.04 + 4.82i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (14.4 - 5.27i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-2.06 + 2.46i)T + (-15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-2.07 + 1.19i)T + (48.5 - 84.0i)T^{2} \) |
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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
−14.20937328017049361791883262640, −12.86725990661226752807246659727, −12.03975432922997926713689429401, −11.31326530344388801655890303292, −9.541237660507926566620426363113, −8.436149927162698828121424636447, −7.53693951191976436890229194528, −6.16899747484674839346908006613, −4.05004939622091891098649431846, −1.23225079956428996471950766790,
3.23913226609077370373622393797, 4.96846762356247204586932777589, 7.00964981948651480842764594671, 7.75007070277758753158486937005, 9.593687216992280868176493881333, 10.20705265829310922244156351758, 11.32732360037952722030788115010, 12.26155199350118035078299889183, 14.26673588998674914824663144252, 15.24902585796246352183882750060
