L(s) = 1 | + (0.767 + 1.32i)5-s + 2.31i·7-s − 5.49i·11-s + (−3.96 − 2.29i)13-s + (2.79 + 4.84i)17-s + (1.09 − 4.21i)19-s + (3.07 + 1.77i)23-s + (1.32 − 2.29i)25-s + (−7.12 − 4.11i)29-s + 6.20·31-s + (−3.07 + 1.77i)35-s − 1.73i·37-s + (5.86 − 3.38i)41-s + (9.30 − 5.37i)43-s + (−1.68 − 0.973i)47-s + ⋯ |
L(s) = 1 | + (0.343 + 0.594i)5-s + 0.874i·7-s − 1.65i·11-s + (−1.10 − 0.635i)13-s + (0.678 + 1.17i)17-s + (0.252 − 0.967i)19-s + (0.641 + 0.370i)23-s + (0.264 − 0.458i)25-s + (−1.32 − 0.764i)29-s + 1.11·31-s + (−0.519 + 0.300i)35-s − 0.284i·37-s + (0.916 − 0.528i)41-s + (1.41 − 0.819i)43-s + (−0.246 − 0.142i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.813039026\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.813039026\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-1.09 + 4.21i)T \) |
good | 5 | \( 1 + (-0.767 - 1.32i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 - 2.31iT - 7T^{2} \) |
| 11 | \( 1 + 5.49iT - 11T^{2} \) |
| 13 | \( 1 + (3.96 + 2.29i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.79 - 4.84i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-3.07 - 1.77i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (7.12 + 4.11i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.20T + 31T^{2} \) |
| 37 | \( 1 + 1.73iT - 37T^{2} \) |
| 41 | \( 1 + (-5.86 + 3.38i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-9.30 + 5.37i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.68 + 0.973i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (8.16 + 4.71i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.76 - 8.24i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.32 - 4.02i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.20 + 7.27i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (7.83 + 13.5i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.79 - 11.7i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.11 - 12.3i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 1.94iT - 83T^{2} \) |
| 89 | \( 1 + (4.82 + 2.78i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.40 + 4.27i)T + (48.5 - 84.0i)T^{2} \) |
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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
−8.819242444008425078193782967336, −8.039439800010335067438646239579, −7.34222350801149329713344566729, −6.27398038138473345502397396710, −5.77141882986858392212054231263, −5.14133990749251000942902382300, −3.82657696210987543366563484153, −2.90293785478793546937009164340, −2.33949793615190602921892978512, −0.69774702160562517015556094323,
1.04566339063805615997116649641, 2.03235088948591581522467347350, 3.15701675091456065551079187624, 4.47314478728593411591371746156, 4.72037395321948235124519541850, 5.64866090049413674879291298775, 6.84492193018989433578801656500, 7.35926212099111669219553216992, 7.85143361550723378580939957222, 9.186004338495087688070516258255