| L(s) = 1 | + (1.92 + 0.0776i)2-s + (−0.278 + 0.960i)3-s + (1.71 + 0.138i)4-s + (3.62 + 1.37i)5-s + (−0.610 + 1.82i)6-s + (1.17 + 2.76i)7-s + (−0.536 − 0.0651i)8-s + (−0.845 − 0.534i)9-s + (6.88 + 2.93i)10-s + (−1.78 − 2.82i)11-s + (−0.609 + 1.60i)12-s + (−2.71 − 2.37i)13-s + (2.05 + 5.42i)14-s + (−2.33 + 3.10i)15-s + (−4.42 − 0.718i)16-s + (−2.15 + 0.918i)17-s + ⋯ |
| L(s) = 1 | + (1.36 + 0.0549i)2-s + (−0.160 + 0.554i)3-s + (0.857 + 0.0691i)4-s + (1.62 + 0.615i)5-s + (−0.249 + 0.746i)6-s + (0.445 + 1.04i)7-s + (−0.189 − 0.0230i)8-s + (−0.281 − 0.178i)9-s + (2.17 + 0.927i)10-s + (−0.538 − 0.851i)11-s + (−0.176 + 0.464i)12-s + (−0.752 − 0.659i)13-s + (0.549 + 1.45i)14-s + (−0.601 + 0.800i)15-s + (−1.10 − 0.179i)16-s + (−0.522 + 0.222i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.579 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.579 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.82715 + 1.45879i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.82715 + 1.45879i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.278 - 0.960i)T \) |
| 13 | \( 1 + (2.71 + 2.37i)T \) |
| good | 2 | \( 1 + (-1.92 - 0.0776i)T + (1.99 + 0.160i)T^{2} \) |
| 5 | \( 1 + (-3.62 - 1.37i)T + (3.74 + 3.31i)T^{2} \) |
| 7 | \( 1 + (-1.17 - 2.76i)T + (-4.84 + 5.04i)T^{2} \) |
| 11 | \( 1 + (1.78 + 2.82i)T + (-4.71 + 9.93i)T^{2} \) |
| 17 | \( 1 + (2.15 - 0.918i)T + (11.7 - 12.2i)T^{2} \) |
| 19 | \( 1 + (-1.03 + 0.596i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.67 + 8.09i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.242 - 6.02i)T + (-28.9 - 2.33i)T^{2} \) |
| 31 | \( 1 + (-0.596 - 0.673i)T + (-3.73 + 30.7i)T^{2} \) |
| 37 | \( 1 + (-9.40 + 1.91i)T + (34.0 - 14.5i)T^{2} \) |
| 41 | \( 1 + (3.14 + 0.911i)T + (34.6 + 21.9i)T^{2} \) |
| 43 | \( 1 + (-1.20 + 5.91i)T + (-39.5 - 16.8i)T^{2} \) |
| 47 | \( 1 + (-0.455 + 0.314i)T + (16.6 - 43.9i)T^{2} \) |
| 53 | \( 1 + (0.0637 - 0.524i)T + (-51.4 - 12.6i)T^{2} \) |
| 59 | \( 1 + (-0.493 - 3.03i)T + (-55.9 + 18.6i)T^{2} \) |
| 61 | \( 1 + (11.0 - 8.27i)T + (16.9 - 58.5i)T^{2} \) |
| 67 | \( 1 + (0.412 + 5.11i)T + (-66.1 + 10.7i)T^{2} \) |
| 71 | \( 1 + (-3.47 + 3.34i)T + (2.85 - 70.9i)T^{2} \) |
| 73 | \( 1 + (7.24 - 13.7i)T + (-41.4 - 60.0i)T^{2} \) |
| 79 | \( 1 + (6.74 + 9.77i)T + (-28.0 + 73.8i)T^{2} \) |
| 83 | \( 1 + (1.18 - 4.79i)T + (-73.4 - 38.5i)T^{2} \) |
| 89 | \( 1 + (2.83 + 1.63i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.01 - 2.46i)T + (19.4 + 95.0i)T^{2} \) |
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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
−10.97660947018334360946695121663, −10.45015977172199239401428130344, −9.290102723559804868637651344958, −8.631105516964525529143235458221, −6.86294749896899544111427704440, −5.90063371614032550528141192849, −5.46240294391148702319555888250, −4.69545981511298652042585277078, −2.94485045227885427465677863873, −2.49487960223716355929726404357,
1.58626527200680112678482977206, 2.64601753095198935963297261587, 4.43375845089897513611517068412, 4.99590678419408460950341757249, 5.89222744352954441900318392441, 6.82438343670001139605101786255, 7.73101133773754826827712268080, 9.304383209206517567263996838797, 9.809988701281211103588573477161, 11.06551210096305654494358999949
