L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + 0.999·6-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.499 + 0.866i)12-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.499 − 0.866i)18-s + (−0.5 − 0.866i)19-s − 0.999·22-s + (−0.499 − 0.866i)24-s + 25-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + 0.999·6-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.499 + 0.866i)12-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.499 − 0.866i)18-s + (−0.5 − 0.866i)19-s − 0.999·22-s + (−0.499 − 0.866i)24-s + 25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6905221807\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6905221807\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 2T + T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 - 2T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.675626250282284301819621274217, −8.872005822859066500471993849773, −8.085165431061078036646098671037, −7.29415095601622746810133358732, −6.70593448228089339048598222995, −5.97709229114389218716498931946, −5.08910701662046257957665643218, −4.18229465631229182325939835204, −2.32222198148900651418149339382, −1.11885187379640775960900927436,
0.969622016610054147839627375538, 2.71750401045953523747993761632, 3.60946115669647138769808295024, 4.40210570371293649090739606281, 5.39435809326929959213658271783, 6.34114175436668172437071567145, 7.47709751847109769312071253867, 8.511850842424088175239556987102, 9.073220083232591533421738958458, 9.849193929429020914930067936236