L(s) = 1 | + (−0.863 + 1.49i)2-s + (−0.490 − 0.849i)4-s + 3.51·5-s − 1.75·8-s + (−3.03 + 5.25i)10-s + 6.09·11-s + (−0.560 + 0.970i)13-s + (2.49 − 4.32i)16-s + (−0.601 + 1.04i)17-s + (−1.10 − 1.90i)19-s + (−1.72 − 2.98i)20-s + (−5.25 + 9.10i)22-s + 1.27·23-s + 7.33·25-s + (−0.967 − 1.67i)26-s + ⋯ |
L(s) = 1 | + (−0.610 + 1.05i)2-s + (−0.245 − 0.424i)4-s + 1.57·5-s − 0.621·8-s + (−0.958 + 1.66i)10-s + 1.83·11-s + (−0.155 + 0.269i)13-s + (0.624 − 1.08i)16-s + (−0.146 + 0.252i)17-s + (−0.252 − 0.438i)19-s + (−0.385 − 0.667i)20-s + (−1.12 + 1.94i)22-s + 0.265·23-s + 1.46·25-s + (−0.189 − 0.328i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0866 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0866 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.758065897\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.758065897\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.863 - 1.49i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 - 3.51T + 5T^{2} \) |
| 11 | \( 1 - 6.09T + 11T^{2} \) |
| 13 | \( 1 + (0.560 - 0.970i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.601 - 1.04i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.10 + 1.90i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 1.27T + 23T^{2} \) |
| 29 | \( 1 + (-3.10 - 5.37i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.0942 + 0.163i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.78 + 3.09i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.68 + 2.91i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.90 + 3.29i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.86 + 4.95i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.16 - 7.22i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.63 - 9.75i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.00 + 10.3i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.95 - 6.85i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 + (2.65 - 4.60i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.60 - 7.98i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.624 - 1.08i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (2.77 + 4.79i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.24 + 14.2i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.472216082020033288102035420483, −9.019700535899582307213670509188, −8.461566441200859350265030102780, −6.99685847864640090869563523185, −6.76056285567733782869732913427, −5.96449103379430663125030665560, −5.20951669863518492273963522557, −3.86824352466801887427306175582, −2.49204918381363744872935851064, −1.27815962758266043552203197539,
1.08874899129151877293826739061, 1.88981987401251061355906093772, 2.82680498301503007001333375211, 3.98025586055326831404215356840, 5.27991576337730961715095030474, 6.30918216674234245311233137140, 6.59479065283043842413187542561, 8.197802902563409737910452425325, 9.053387462073202406708165529153, 9.623018673049777530862084610885