L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (0.500 − 0.866i)4-s + (1 + 1.73i)5-s − 0.999·6-s + 3·8-s + (−0.499 − 0.866i)9-s + (−0.999 + 1.73i)10-s + (−2 + 3.46i)11-s + (0.499 + 0.866i)12-s − 2·13-s − 1.99·15-s + (0.500 + 0.866i)16-s + (3 − 5.19i)17-s + (0.499 − 0.866i)18-s + (−2 − 3.46i)19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (0.250 − 0.433i)4-s + (0.447 + 0.774i)5-s − 0.408·6-s + 1.06·8-s + (−0.166 − 0.288i)9-s + (−0.316 + 0.547i)10-s + (−0.603 + 1.04i)11-s + (0.144 + 0.249i)12-s − 0.554·13-s − 0.516·15-s + (0.125 + 0.216i)16-s + (0.727 − 1.26i)17-s + (0.117 − 0.204i)18-s + (−0.458 − 0.794i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.13573 + 0.755468i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.13573 + 0.755468i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3 + 5.19i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-3 + 5.19i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + (-7 - 12.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 18T + 97T^{2} \) |
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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
−13.56775609668087760755850906591, −12.24779454433049900567752070553, −10.99352499726750975058624035967, −10.24219156000314307007322760173, −9.457711587475664148183170594594, −7.50422561668344633323687756596, −6.77557955340642932004863758125, −5.52741282057284022275560102929, −4.63457634943671944158429689057, −2.52632112391922761361217206543,
1.73281414074435367102549418908, 3.40825931087961282193072698863, 5.03255330496125635583567660679, 6.18865728835629433184375431217, 7.73774369923175224258379986051, 8.510065100013018042821434896286, 10.11841087951395740715547439771, 11.03316644612270975077018765656, 12.10556268985430732450527878656, 12.80677120594093965503210946899