L(s) = 1 | + (−1.41 + 0.816i)2-s + (0.334 − 0.579i)4-s + (0.437 − 0.252i)7-s − 2.17i·8-s + (1.55 + 2.68i)11-s + (−5.40 − 3.11i)13-s + (−0.412 + 0.714i)14-s + (2.44 + 4.23i)16-s + 6.10i·17-s + 5.57·19-s + (−4.38 − 2.53i)22-s + (3.31 + 1.91i)23-s + 10.1·26-s − 0.338i·28-s + (−1.22 − 2.12i)29-s + ⋯ |
L(s) = 1 | + (−1.00 + 0.577i)2-s + (0.167 − 0.289i)4-s + (0.165 − 0.0955i)7-s − 0.768i·8-s + (0.467 + 0.809i)11-s + (−1.49 − 0.865i)13-s + (−0.110 + 0.191i)14-s + (0.611 + 1.05i)16-s + 1.47i·17-s + 1.27·19-s + (−0.935 − 0.539i)22-s + (0.690 + 0.398i)23-s + 1.99·26-s − 0.0638i·28-s + (−0.228 − 0.395i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.581 - 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.301862 + 0.586427i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.301862 + 0.586427i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (1.41 - 0.816i)T + (1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (-0.437 + 0.252i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.55 - 2.68i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (5.40 + 3.11i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 6.10iT - 17T^{2} \) |
| 19 | \( 1 - 5.57T + 19T^{2} \) |
| 23 | \( 1 + (-3.31 - 1.91i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.22 + 2.12i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.11 - 3.66i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 6.72iT - 37T^{2} \) |
| 41 | \( 1 + (2.72 - 4.71i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.14 - 0.663i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.21 + 1.85i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 2.54iT - 53T^{2} \) |
| 59 | \( 1 + (1.44 - 2.49i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.42 - 2.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.08 + 1.20i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 5.54T + 71T^{2} \) |
| 73 | \( 1 - 11.7iT - 73T^{2} \) |
| 79 | \( 1 + (-1.70 - 2.94i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (12.0 - 6.95i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 3.38T + 89T^{2} \) |
| 97 | \( 1 + (-9.59 + 5.53i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.30088799830029462208919466774, −9.880903176381747736733497043913, −9.070947874663839858344515262148, −8.074366536818859354283976383480, −7.46417923699309022795171383977, −6.75611240458946854791342642152, −5.53121763689443542206929509996, −4.41833407786083777841320377351, −3.13386073950004639568547660040, −1.37954686040797182543983595507,
0.53084517113710721138178702877, 2.04339525537422877577786067471, 3.14100505308249705597795503741, 4.77144794279045347655594843590, 5.52671153339735832829912895287, 7.02156845914993525932074332921, 7.64771495281672630151944506025, 8.910347411268418729922572641395, 9.299265102088723652829147318359, 10.00667746083049600195074626938