L(s) = 1 | + (−1.68 + 2.92i)5-s + (2.35 + 4.07i)7-s + (−0.437 − 0.758i)11-s + (0.686 − 1.18i)13-s + 2.37·17-s − 5.57·19-s + (−2.35 + 4.07i)23-s + (−3.18 − 5.51i)25-s + (−2.68 − 4.65i)29-s + (−3.22 + 5.58i)31-s − 15.8·35-s + 4·37-s + (0.5 − 0.866i)41-s + (−0.437 − 0.758i)43-s + (−2.35 − 4.07i)47-s + ⋯ |
L(s) = 1 | + (−0.754 + 1.30i)5-s + (0.888 + 1.53i)7-s + (−0.131 − 0.228i)11-s + (0.190 − 0.329i)13-s + 0.575·17-s − 1.27·19-s + (−0.490 + 0.849i)23-s + (−0.637 − 1.10i)25-s + (−0.498 − 0.863i)29-s + (−0.579 + 1.00i)31-s − 2.68·35-s + 0.657·37-s + (0.0780 − 0.135i)41-s + (−0.0667 − 0.115i)43-s + (−0.342 − 0.594i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.404278 + 1.06397i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.404278 + 1.06397i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.68 - 2.92i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-2.35 - 4.07i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.437 + 0.758i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.686 + 1.18i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 2.37T + 17T^{2} \) |
| 19 | \( 1 + 5.57T + 19T^{2} \) |
| 23 | \( 1 + (2.35 - 4.07i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.68 + 4.65i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.22 - 5.58i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.437 + 0.758i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.35 + 4.07i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + (4.26 - 7.38i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.05 - 1.83i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.26 - 7.38i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 9.40T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + (-3.22 - 5.58i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.47 - 2.55i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + (4.5 + 7.79i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.73529169145704625588483546578, −9.687573163414936534362882238786, −8.561623755513898470637624543206, −8.048048856581465629566506372471, −7.15837329111823087578294901367, −6.07415390751696565206213757607, −5.38705356539869237390138357585, −4.04876946109379719096738563259, −2.99865591782512081783738525262, −2.04780588851623670655966269351,
0.55465846190789714065312651307, 1.75519299472058254708993608345, 3.81843820351628821543573110685, 4.37872050402623136491327424353, 5.07907395534867680560046010289, 6.46757194637334444603156876257, 7.60950712636972750540286049511, 8.023224578791604230555685405232, 8.835143391196431454244696286114, 9.853392514718994165727954253571