L(s) = 1 | + (−1.35 + 2.35i)2-s + (−2.68 − 4.65i)4-s + 1.58·5-s + 9.15·8-s + (−2.15 + 3.73i)10-s + 1.34·11-s + (1.58 − 2.75i)13-s + (−7.05 + 12.2i)16-s + (1.40 − 2.42i)17-s + (−0.312 − 0.541i)19-s + (−4.26 − 7.38i)20-s + (−1.83 + 3.17i)22-s + 0.284·23-s − 2.48·25-s + (4.31 + 7.47i)26-s + ⋯ |
L(s) = 1 | + (−0.959 + 1.66i)2-s + (−1.34 − 2.32i)4-s + 0.709·5-s + 3.23·8-s + (−0.681 + 1.17i)10-s + 0.406·11-s + (0.440 − 0.763i)13-s + (−1.76 + 3.05i)16-s + (0.339 − 0.588i)17-s + (−0.0717 − 0.124i)19-s + (−0.952 − 1.65i)20-s + (−0.390 + 0.676i)22-s + 0.0593·23-s − 0.496·25-s + (0.846 + 1.46i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8920918921\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8920918921\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.35 - 2.35i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 - 1.58T + 5T^{2} \) |
| 11 | \( 1 - 1.34T + 11T^{2} \) |
| 13 | \( 1 + (-1.58 + 2.75i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.40 + 2.42i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.312 + 0.541i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 0.284T + 23T^{2} \) |
| 29 | \( 1 + (2.27 + 3.93i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.71 + 6.43i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.01 + 6.94i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.01 - 8.68i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.12 + 5.42i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.57 + 9.65i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.39 + 2.41i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.28 - 3.96i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.192 + 0.333i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.26 - 2.19i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 1.45T + 71T^{2} \) |
| 73 | \( 1 + (-0.234 + 0.405i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.85 + 13.6i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.99 - 12.1i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1.29 + 2.24i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.22 - 12.5i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.465417225957140188804554269171, −8.788988187028920748899080208164, −7.971828097770756765156841423165, −7.28473226141918345630602763822, −6.44566165221086187092372190652, −5.70281957262685104846156174204, −5.19495881851061369890867933968, −3.87410497300994343249191007150, −1.95644630557693991150651589408, −0.55574574804268292425326476360,
1.35076763420659789725733554150, 1.97943294727964566246097478280, 3.24720288693353955682497884711, 3.97061386944087834012035992374, 5.12074539032295755948056879921, 6.43347651516657076783975494620, 7.46081821652526103060523490007, 8.483000289914366045436945124475, 9.009234451627173536322121404098, 9.679203654176516501595990018489