L(s) = 1 | + (0.5 − 0.866i)3-s + (1.5 − 0.866i)5-s + (−0.5 − 2.59i)7-s + (−0.499 − 0.866i)9-s + (−1.5 − 0.866i)11-s − 1.73i·15-s + (3 + 1.73i)17-s + (−1 − 1.73i)19-s + (−2.5 − 0.866i)21-s + (−1 + 1.73i)25-s − 0.999·27-s + 9·29-s + (2.5 − 4.33i)31-s + (−1.5 + 0.866i)33-s + (−3 − 3.46i)35-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (0.670 − 0.387i)5-s + (−0.188 − 0.981i)7-s + (−0.166 − 0.288i)9-s + (−0.452 − 0.261i)11-s − 0.447i·15-s + (0.727 + 0.420i)17-s + (−0.229 − 0.397i)19-s + (−0.545 − 0.188i)21-s + (−0.200 + 0.346i)25-s − 0.192·27-s + 1.67·29-s + (0.449 − 0.777i)31-s + (−0.261 + 0.150i)33-s + (−0.507 − 0.585i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.23071 - 0.891612i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23071 - 0.891612i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 + 2.59i)T \) |
good | 5 | \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 + 0.866i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 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 | \( 1 + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5 + 8.66i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 10.3iT - 41T^{2} \) |
| 43 | \( 1 - 3.46iT - 43T^{2} \) |
| 47 | \( 1 + (-6 - 10.3i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-12 - 6.92i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 13.8iT - 71T^{2} \) |
| 73 | \( 1 + (6 + 3.46i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.5 - 2.59i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3T + 83T^{2} \) |
| 89 | \( 1 + (3 - 1.73i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 19.0iT - 97T^{2} \) |
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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
−11.33807669996562159470952058435, −10.32607441806919193274367833830, −9.594306911347874855998039066891, −8.458415058171009205841618221691, −7.60754302499055344656968711218, −6.56596142331219444038537919859, −5.55437970291621167158891736823, −4.20967410185630123000259716896, −2.78302971185410998715134721106, −1.15085164351011270954554162308,
2.22489091886172336378924877710, 3.25300767388092965948078299974, 4.87693053333508069501427701415, 5.77456021823975596808470790083, 6.83417513101135604906306015068, 8.200417466545870987154485219810, 8.975605550546403207334380693422, 10.09236921106327671604554299848, 10.39770519244131540883513627285, 11.89283928389605238915491261049