L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−1.23 + 2.13i)5-s + (−1.03 + 2.43i)7-s + 0.999i·8-s − 2.46i·10-s + (−2.27 + 1.31i)11-s + (−2.35 − 1.35i)13-s + (−0.320 − 2.62i)14-s + (−0.5 − 0.866i)16-s − 4.00·17-s − 8.27i·19-s + (1.23 + 2.13i)20-s + (1.31 − 2.27i)22-s + (1.39 + 0.807i)23-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.550 + 0.953i)5-s + (−0.391 + 0.920i)7-s + 0.353i·8-s − 0.778i·10-s + (−0.686 + 0.396i)11-s + (−0.652 − 0.376i)13-s + (−0.0856 − 0.701i)14-s + (−0.125 − 0.216i)16-s − 0.972·17-s − 1.89i·19-s + (0.275 + 0.476i)20-s + (0.280 − 0.485i)22-s + (0.291 + 0.168i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0341 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0341 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1624783875\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1624783875\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.03 - 2.43i)T \) |
good | 5 | \( 1 + (1.23 - 2.13i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.27 - 1.31i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.35 + 1.35i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 4.00T + 17T^{2} \) |
| 19 | \( 1 + 8.27iT - 19T^{2} \) |
| 23 | \( 1 + (-1.39 - 0.807i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.71 - 0.992i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.737 - 0.425i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 + (-2.76 + 4.78i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.61 + 6.25i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.90 - 10.2i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 8.19iT - 53T^{2} \) |
| 59 | \( 1 + (-0.773 + 1.34i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7.57 - 4.37i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.230 + 0.399i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 7.30iT - 71T^{2} \) |
| 73 | \( 1 + 0.0359iT - 73T^{2} \) |
| 79 | \( 1 + (0.700 + 1.21i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (8.06 + 13.9i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + (-1.56 + 0.901i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.419008111981854603481231481013, −8.935697194438760906454090021440, −7.83157945091050343876151985245, −7.17643270161107768343384878412, −6.53411749401828295059341293775, −5.47887457193378915418128456801, −4.53522445243372996475300474879, −2.92253485687050346538547500378, −2.40947099092654019930607663200, −0.094195536625100030746116560419,
1.17645882069965356281134362815, 2.65693510659582357931605233456, 3.95674818709583499285380584515, 4.54228322470897445113603648951, 5.85023815370095869461659658536, 6.91001578343995826843483551919, 7.88082830237737194521554447878, 8.236252007336315906702700515532, 9.306414115975422190632230903253, 9.938073016761964709618681031916