L(s) = 1 | + (0.5 + 0.866i)2-s + (1.18 + 1.26i)3-s + (−0.499 + 0.866i)4-s + (0.686 − 1.18i)5-s + (−0.5 + 1.65i)6-s + (−0.5 − 0.866i)7-s − 0.999·8-s + (−0.186 + 2.99i)9-s + 1.37·10-s + (−2.18 − 3.78i)11-s + (−1.68 + 0.396i)12-s + (−1 + 1.73i)13-s + (0.499 − 0.866i)14-s + (2.31 − 0.543i)15-s + (−0.5 − 0.866i)16-s − 4.37·17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.684 + 0.728i)3-s + (−0.249 + 0.433i)4-s + (0.306 − 0.531i)5-s + (−0.204 + 0.677i)6-s + (−0.188 − 0.327i)7-s − 0.353·8-s + (−0.0620 + 0.998i)9-s + 0.433·10-s + (−0.659 − 1.14i)11-s + (−0.486 + 0.114i)12-s + (−0.277 + 0.480i)13-s + (0.133 − 0.231i)14-s + (0.597 − 0.140i)15-s + (−0.125 − 0.216i)16-s − 1.06·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.21916 + 0.798450i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.21916 + 0.798450i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.18 - 1.26i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-0.686 + 1.18i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.18 + 3.78i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 4.37T + 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + (-3.68 + 6.38i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.37 + 2.37i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (5.18 - 8.98i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.55 + 7.89i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 2.74T + 53T^{2} \) |
| 59 | \( 1 + (3.55 - 6.16i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.05 - 12.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.55 - 13.0i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 + 5.11T + 73T^{2} \) |
| 79 | \( 1 + (6.05 + 10.4i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.74 - 4.75i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 3.25T + 89T^{2} \) |
| 97 | \( 1 + (4.55 + 7.89i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−13.52800061268005924137919066798, −13.15673273802718201535620283140, −11.47021211354963289362890518497, −10.30125430188152363037732730294, −9.087993171069103384190256023281, −8.418965819437021315515620696375, −7.07303048009666643407173618507, −5.50179419082868554959466102205, −4.46452392562003441426142369778, −2.98073045426211311221302203718,
2.10732594047403510097966828064, 3.24916221363599159764289039433, 5.15364717351957514406832532988, 6.67740331843246986413126198964, 7.69647613661721236000456706668, 9.185892151627262300803817088652, 9.998694303896251120023691501641, 11.29375485244275901812657898716, 12.41835551712099721948875087964, 13.12866599705885963758926237063