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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2358634624\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2358634624\) |
\(L(\frac{3}{2})\) |
|
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
−9.203870248299821737910318433164, −8.672058553857247378806950843186, −7.83995868395391111487377015914, −7.22683411042739309695997496032, −6.51900334918083076383527889062, −5.70959100722992944345412054087, −4.69631760313283328104129012846, −3.89227568146268560568049834356, −1.76028931475236001170397513945, −0.16221893030479343460366099650,
1.10327951750280438663869569257, 2.44696029002176707734253414659, 3.29580445552573768104805453942, 4.15232262511722069032555843914, 5.06312379694768031819924530038, 6.74960819862968928951165153646, 7.81208960735465246880410374439, 8.165562442077159731533692685250, 9.166776592258802777271028951292, 9.764792419591640312387910348011