| L(s) = 1 | + (−0.358 − 0.933i)2-s + (−1.59 − 2.44i)3-s + (−0.743 + 0.669i)4-s + (−1.02 − 1.98i)5-s + (−1.71 + 2.36i)6-s + (1.81 − 1.92i)7-s + (0.891 + 0.453i)8-s + (−2.24 + 5.04i)9-s + (−1.48 + 1.66i)10-s + (0.393 + 3.29i)11-s + (2.82 + 0.755i)12-s + (−4.47 − 0.709i)13-s + (−2.44 − 1.01i)14-s + (−3.24 + 5.66i)15-s + (0.104 − 0.994i)16-s + (0.807 − 2.10i)17-s + ⋯ |
| L(s) = 1 | + (−0.253 − 0.660i)2-s + (−0.918 − 1.41i)3-s + (−0.371 + 0.334i)4-s + (−0.457 − 0.889i)5-s + (−0.700 + 0.964i)6-s + (0.687 − 0.725i)7-s + (0.315 + 0.160i)8-s + (−0.749 + 1.68i)9-s + (−0.470 + 0.527i)10-s + (0.118 + 0.992i)11-s + (0.814 + 0.218i)12-s + (−1.24 − 0.196i)13-s + (−0.653 − 0.270i)14-s + (−0.836 + 1.46i)15-s + (0.0261 − 0.248i)16-s + (0.195 − 0.510i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.653 - 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.653 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0998190 + 0.0457152i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0998190 + 0.0457152i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.358 + 0.933i)T \) |
| 5 | \( 1 + (1.02 + 1.98i)T \) |
| 7 | \( 1 + (-1.81 + 1.92i)T \) |
| 11 | \( 1 + (-0.393 - 3.29i)T \) |
| good | 3 | \( 1 + (1.59 + 2.44i)T + (-1.22 + 2.74i)T^{2} \) |
| 13 | \( 1 + (4.47 + 0.709i)T + (12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-0.807 + 2.10i)T + (-12.6 - 11.3i)T^{2} \) |
| 19 | \( 1 + (4.51 - 5.01i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (7.51 + 2.01i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-3.24 - 1.05i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-8.33 + 0.876i)T + (30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-3.54 - 2.30i)T + (15.0 + 33.8i)T^{2} \) |
| 41 | \( 1 + (-0.464 + 0.150i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-0.127 + 0.127i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.353 + 6.74i)T + (-46.7 + 4.91i)T^{2} \) |
| 53 | \( 1 + (-5.87 - 4.75i)T + (11.0 + 51.8i)T^{2} \) |
| 59 | \( 1 + (5.40 + 5.99i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (7.24 + 0.761i)T + (59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (14.5 - 3.90i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (5.22 + 3.79i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (0.497 - 9.49i)T + (-72.6 - 7.63i)T^{2} \) |
| 79 | \( 1 + (-2.00 + 4.51i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-0.126 - 0.801i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (6.87 + 11.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.62 - 10.2i)T + (-92.2 - 29.9i)T^{2} \) |
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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
−9.900701451501440158925296047125, −8.421224023666524998676264965812, −7.79697127946084346216982779695, −7.25582047069205707202793997201, −6.09040329193476437307204749960, −4.85844369558676265617780061358, −4.29664934217519047547475227883, −2.22827858140807544734578600144, −1.31001198750668581204114377084, −0.07215584293505019293160860596,
2.72350908449748329740357248939, 4.17227711957758281044491733753, 4.75931024467020036808876260620, 5.89796387143020655752780541805, 6.35049366012174077835944815531, 7.68545665652834913322831873371, 8.537513626663055705117651842965, 9.441843009991229414436424658108, 10.29941899212179349299070572406, 10.83963652578321338793215269818
