L(s) = 1 | + (0.5 − 0.866i)3-s + (1.5 − 0.866i)5-s + (−2.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (4.5 + 2.59i)11-s + 6.92i·13-s − 1.73i·15-s + (3 + 1.73i)17-s + (−1 − 1.73i)19-s + (−0.500 + 2.59i)21-s + (−6 + 3.46i)23-s + (−1 + 1.73i)25-s − 0.999·27-s + 9·29-s + (0.5 − 0.866i)31-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (0.670 − 0.387i)5-s + (−0.944 + 0.327i)7-s + (−0.166 − 0.288i)9-s + (1.35 + 0.783i)11-s + 1.92i·13-s − 0.447i·15-s + (0.727 + 0.420i)17-s + (−0.229 − 0.397i)19-s + (−0.109 + 0.566i)21-s + (−1.25 + 0.722i)23-s + (−0.200 + 0.346i)25-s − 0.192·27-s + 1.67·29-s + (0.0898 − 0.155i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.915958458\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.915958458\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (2.5 - 0.866i)T \) |
good | 5 | \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-4.5 - 2.59i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 6.92iT - 13T^{2} \) |
| 17 | \( 1 + (-3 - 1.73i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (6 - 3.46i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 3.46iT - 43T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6 + 3.46i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6.92iT - 71T^{2} \) |
| 73 | \( 1 + (-6 - 3.46i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.5 + 0.866i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 15T + 83T^{2} \) |
| 89 | \( 1 + (9 - 5.19i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 8.66iT - 97T^{2} \) |
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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
−9.535065975846172405240958564469, −9.087756639684113404695772944098, −8.173762841779971970736320018325, −6.87261802571745326498665855103, −6.59305759877207047667008374207, −5.73708288278090374630242645945, −4.43509131294369892277703701017, −3.62173566644293999277133840253, −2.20306919143193019036465934251, −1.44498877374486520915247016129,
0.819391462600540293555126408636, 2.63443780697404757407179083800, 3.35279970094034003351239438763, 4.20264034259411299963558811676, 5.64802043621953387760691101209, 6.09462272215053025909891215494, 6.98180667752108816020964141101, 8.144975251171845251372689290064, 8.734897173170311866222437243811, 9.901703733266895180542999915957