L(s) = 1 | − 2-s + (−0.5 − 0.866i)3-s + 4-s + (0.5 + 0.866i)6-s − 8-s + (−0.499 + 0.866i)9-s + (1.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + 16-s + (−1.5 + 0.866i)17-s + (0.499 − 0.866i)18-s + (0.5 + 0.866i)19-s + (−1.5 − 0.866i)22-s + (0.5 + 0.866i)24-s + (0.5 + 0.866i)25-s + ⋯ |
L(s) = 1 | − 2-s + (−0.5 − 0.866i)3-s + 4-s + (0.5 + 0.866i)6-s − 8-s + (−0.499 + 0.866i)9-s + (1.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + 16-s + (−1.5 + 0.866i)17-s + (0.499 − 0.866i)18-s + (0.5 + 0.866i)19-s + (−1.5 − 0.866i)22-s + (0.5 + 0.866i)24-s + (0.5 + 0.866i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5839558076\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5839558076\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - 1.73iT - T^{2} \) |
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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
−9.721355463959134691161741118030, −8.969248259023872714596308549739, −8.206131462325004199271734023771, −7.36220772233599304417868007460, −6.58723751399258118460112075738, −6.23629704951720967258355848582, −4.93378302681367424376007331369, −3.59988044271356916795739855126, −2.08902770502707213204060737146, −1.37413255242098916729067087745,
0.793951325935148882853601502898, 2.55606483507367598836196841208, 3.64975772518846747556985400865, 4.66502369776871945739365329030, 5.81754995497184410893113815522, 6.58003573195336483460411007912, 7.18135705888495785062953160495, 8.752769495195393819796676600604, 8.860147512753636417823836705957, 9.610367479125074542375809165438