| L(s) = 1 | + (−0.884 + 0.467i)2-s + (1.04 + 2.38i)3-s + (0.563 − 0.826i)4-s + (−1.44 + 1.70i)5-s + (−2.03 − 1.62i)6-s + (−2.46 − 0.960i)7-s + (−0.111 + 0.993i)8-s + (−2.56 + 2.76i)9-s + (0.482 − 2.18i)10-s + (0.342 − 0.318i)11-s + (2.55 + 0.483i)12-s + (−3.39 − 2.13i)13-s + (2.62 − 0.303i)14-s + (−5.57 − 1.67i)15-s + (−0.365 − 0.930i)16-s + (−1.97 + 2.29i)17-s + ⋯ |
| L(s) = 1 | + (−0.625 + 0.330i)2-s + (0.601 + 1.37i)3-s + (0.281 − 0.413i)4-s + (−0.647 + 0.762i)5-s + (−0.830 − 0.662i)6-s + (−0.931 − 0.362i)7-s + (−0.0395 + 0.351i)8-s + (−0.856 + 0.922i)9-s + (0.152 − 0.690i)10-s + (0.103 − 0.0959i)11-s + (0.738 + 0.139i)12-s + (−0.941 − 0.591i)13-s + (0.702 − 0.0810i)14-s + (−1.43 − 0.433i)15-s + (−0.0913 − 0.232i)16-s + (−0.479 + 0.557i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.165142 - 0.453855i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.165142 - 0.453855i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.884 - 0.467i)T \) |
| 5 | \( 1 + (1.44 - 1.70i)T \) |
| 7 | \( 1 + (2.46 + 0.960i)T \) |
| good | 3 | \( 1 + (-1.04 - 2.38i)T + (-2.04 + 2.19i)T^{2} \) |
| 11 | \( 1 + (-0.342 + 0.318i)T + (0.822 - 10.9i)T^{2} \) |
| 13 | \( 1 + (3.39 + 2.13i)T + (5.64 + 11.7i)T^{2} \) |
| 17 | \( 1 + (1.97 - 2.29i)T + (-2.53 - 16.8i)T^{2} \) |
| 19 | \( 1 + (0.173 - 0.299i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.25 - 1.93i)T + (3.42 - 22.7i)T^{2} \) |
| 29 | \( 1 + (1.26 - 2.62i)T + (-18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (-2.53 + 1.46i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.63 - 8.61i)T + (-34.4 - 13.5i)T^{2} \) |
| 41 | \( 1 + (-3.84 + 3.06i)T + (9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (4.68 - 0.527i)T + (41.9 - 9.56i)T^{2} \) |
| 47 | \( 1 + (-4.10 - 7.76i)T + (-26.4 + 38.8i)T^{2} \) |
| 53 | \( 1 + (1.72 + 9.12i)T + (-49.3 + 19.3i)T^{2} \) |
| 59 | \( 1 + (0.345 + 0.0521i)T + (56.3 + 17.3i)T^{2} \) |
| 61 | \( 1 + (2.93 + 4.30i)T + (-22.2 + 56.7i)T^{2} \) |
| 67 | \( 1 + (9.08 + 2.43i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (11.8 - 5.69i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (6.40 - 12.1i)T + (-41.1 - 60.3i)T^{2} \) |
| 79 | \( 1 + (13.7 + 7.91i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.39 - 5.40i)T + (-36.0 + 74.7i)T^{2} \) |
| 89 | \( 1 + (-12.7 - 11.8i)T + (6.65 + 88.7i)T^{2} \) |
| 97 | \( 1 + (-10.4 - 10.4i)T + 97iT^{2} \) |
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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
−11.04435925916197447892662853697, −10.23663712863609567637718052055, −9.913913712137613269463275613098, −8.950031637599982278866340469045, −8.021486033274662630536879416871, −7.12363458446223729846134088213, −6.09448382969890532650680189878, −4.65591197575430871984214664447, −3.62227886558555890813722573015, −2.77311116924381459644876741738,
0.31237407563023448731689550238, 1.93224596073907269794808923974, 2.96227503220154683408985656503, 4.41450895534577355494282283454, 6.09076215991854120848247348574, 7.17859907037458337954892488161, 7.59815316279733812787217867519, 8.808713692425502773453223987408, 9.100737492813338899611966547074, 10.25809645774698779183425238737
