L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s − 3.27·5-s + (−0.499 + 0.866i)6-s + (−0.5 + 0.866i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (1.63 + 2.83i)10-s + (2 + 3.46i)11-s + 0.999·12-s + (3.5 − 0.866i)13-s + 0.999·14-s + (1.63 + 2.83i)15-s + (−0.5 − 0.866i)16-s + (−1.5 + 2.59i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s − 1.46·5-s + (−0.204 + 0.353i)6-s + (−0.188 + 0.327i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.517 + 0.896i)10-s + (0.603 + 1.04i)11-s + 0.288·12-s + (0.970 − 0.240i)13-s + 0.267·14-s + (0.422 + 0.732i)15-s + (−0.125 − 0.216i)16-s + (−0.363 + 0.630i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.761390 - 0.209255i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.761390 - 0.209255i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (-3.5 + 0.866i)T \) |
good | 5 | \( 1 + 3.27T + 5T^{2} \) |
| 11 | \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.27 + 7.40i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.13 + 1.97i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.362 + 0.627i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.27T + 31T^{2} \) |
| 37 | \( 1 + (-3.63 - 6.30i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.362 + 0.627i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.41 - 9.37i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 8.54T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + (5.41 - 9.37i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.5 + 7.79i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.13 - 8.89i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.13 + 1.97i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + (4.13 + 7.16i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.27 - 12.6i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.03712851761541453478982750309, −9.916071634235912576587484597276, −8.848196750661737564426410476772, −8.144325962756864746498641701013, −7.23973489423363799615641590840, −6.42328422609142078290890984412, −4.79920863205867065813416041662, −3.91698530874514036110549935709, −2.68899801264906544670085670522, −0.972198847389114395569722339515,
0.77980177529961547556015709261, 3.58636921122831018395502785470, 3.99861042369625814833989955675, 5.42282855183347354190328533060, 6.36180696536628095206601972150, 7.38474402246854313751929396183, 8.214573436989799727686142794745, 8.919060818049423603908099878907, 9.957039420396949359889835836421, 10.93984887230098916820572081510