L(s) = 1 | + (1.08 + 1.88i)2-s + (−1.36 + 2.36i)4-s + 1.26·5-s − 1.60·8-s + (1.38 + 2.39i)10-s + 5.47·11-s + (−2.37 − 4.10i)13-s + (0.992 + 1.71i)16-s + (2.40 + 4.17i)17-s + (2.69 − 4.66i)19-s + (−1.73 + 3.00i)20-s + (5.96 + 10.3i)22-s + 5.17·23-s − 3.39·25-s + (5.16 − 8.94i)26-s + ⋯ |
L(s) = 1 | + (0.769 + 1.33i)2-s + (−0.684 + 1.18i)4-s + 0.567·5-s − 0.566·8-s + (0.436 + 0.755i)10-s + 1.65·11-s + (−0.658 − 1.13i)13-s + (0.248 + 0.429i)16-s + (0.584 + 1.01i)17-s + (0.617 − 1.06i)19-s + (−0.388 + 0.672i)20-s + (1.27 + 2.20i)22-s + 1.07·23-s − 0.678·25-s + (1.01 − 1.75i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.248 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.065999679\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.065999679\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1.08 - 1.88i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 - 1.26T + 5T^{2} \) |
| 11 | \( 1 - 5.47T + 11T^{2} \) |
| 13 | \( 1 + (2.37 + 4.10i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.40 - 4.17i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.69 + 4.66i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 5.17T + 23T^{2} \) |
| 29 | \( 1 + (2.01 - 3.49i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.732 - 1.26i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.959 - 1.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.94 + 3.37i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.66 - 2.87i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.57 + 2.73i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.57 + 6.18i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.154 - 0.267i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.17 - 8.95i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.23 - 3.87i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 1.96T + 71T^{2} \) |
| 73 | \( 1 + (5.27 + 9.13i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.50 - 7.80i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.08 - 8.79i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.59 - 4.49i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.48 + 4.30i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.672179554808409051971381748516, −8.877078107392054412054282372231, −8.030641657866522251477785796222, −7.09364087471461955971457089607, −6.61401433843326927705341356696, −5.63723466483484715236127209472, −5.16079492211994356987173184907, −4.06107726738625664994664432119, −3.14714295289295833681114716473, −1.42488083371382360546740838649,
1.25556101641133894960111877596, 2.06436060697981782381457766911, 3.23900192777405599350182190353, 4.04670105679061807858664797652, 4.88830246610828380926356784011, 5.80969076411766073168697340806, 6.79406309505078080726329884744, 7.68654453166767928595401806133, 9.233294507593689775377490102865, 9.486083690511389516776116301343