L(s) = 1 | + (1.36 − 2.36i)3-s + (0.702 + 2.62i)5-s + (2.97 + 1.71i)7-s + (−2.23 − 3.86i)9-s + 0.327·11-s + (1.36 + 5.09i)13-s + (7.15 + 1.91i)15-s + (−2.97 − 0.795i)17-s + (3.12 − 0.836i)19-s + (8.12 − 4.68i)21-s + (0.136 + 0.136i)23-s + (−2.04 + 1.17i)25-s − 4.00·27-s + (−5.38 − 5.38i)29-s + (−5.98 + 5.98i)31-s + ⋯ |
L(s) = 1 | + (0.788 − 1.36i)3-s + (0.314 + 1.17i)5-s + (1.12 + 0.648i)7-s + (−0.744 − 1.28i)9-s + 0.0988·11-s + (0.378 + 1.41i)13-s + (1.84 + 0.495i)15-s + (−0.720 − 0.193i)17-s + (0.715 − 0.191i)19-s + (1.77 − 1.02i)21-s + (0.0284 + 0.0284i)23-s + (−0.408 + 0.235i)25-s − 0.769·27-s + (−0.999 − 0.999i)29-s + (−1.07 + 1.07i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.17972 - 0.296437i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.17972 - 0.296437i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 37 | \( 1 + (-1.81 + 5.80i)T \) |
good | 3 | \( 1 + (-1.36 + 2.36i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.702 - 2.62i)T + (-4.33 + 2.5i)T^{2} \) |
| 7 | \( 1 + (-2.97 - 1.71i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 0.327T + 11T^{2} \) |
| 13 | \( 1 + (-1.36 - 5.09i)T + (-11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + (2.97 + 0.795i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-3.12 + 0.836i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-0.136 - 0.136i)T + 23iT^{2} \) |
| 29 | \( 1 + (5.38 + 5.38i)T + 29iT^{2} \) |
| 31 | \( 1 + (5.98 - 5.98i)T - 31iT^{2} \) |
| 41 | \( 1 + (-1.5 - 0.866i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.88 + 4.88i)T + 43iT^{2} \) |
| 47 | \( 1 + 10.1iT - 47T^{2} \) |
| 53 | \( 1 + (5.43 + 9.40i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.55 + 9.52i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (9.67 - 2.59i)T + (52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (-3.16 + 5.48i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-10.5 - 6.10i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 9.81iT - 73T^{2} \) |
| 79 | \( 1 + (0.649 - 0.174i)T + (68.4 - 39.5i)T^{2} \) |
| 83 | \( 1 + (-9.32 + 5.38i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.63 - 6.11i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (12.0 - 12.0i)T - 97iT^{2} \) |
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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
−10.99551864169415273261980445094, −9.492631677240916318936466337484, −8.760403006348721127876215765007, −7.913547642220964679060336313636, −6.99306647440828933754968312167, −6.56446925858774087698203259559, −5.27558329235198678742200878778, −3.65320230839467900014950109229, −2.29536797237329172348410668652, −1.82072442600759670035717607645,
1.40937360188973429026923125010, 3.15016753343548609965206329850, 4.25499272030446974402512788061, 4.89866453906961732066738786717, 5.72081826095323839072109294287, 7.64680852133349581930039584001, 8.240254045164234762752675512541, 9.084506295316098104723028888049, 9.655206174076302622238710475599, 10.73516291534673844899577554848