L(s) = 1 | + (0.5 + 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.499 + 0.866i)4-s + 1.73·5-s + (0.866 + 0.499i)6-s − 0.999·8-s + (0.499 − 0.866i)9-s + (0.866 + 1.49i)10-s + 0.999i·12-s + (1.49 − 0.866i)15-s + (−0.5 − 0.866i)16-s + 0.999·18-s + (−0.866 + 1.5i)19-s + (−0.866 + 1.49i)20-s − 23-s + (−0.866 + 0.5i)24-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.499 + 0.866i)4-s + 1.73·5-s + (0.866 + 0.499i)6-s − 0.999·8-s + (0.499 − 0.866i)9-s + (0.866 + 1.49i)10-s + 0.999i·12-s + (1.49 − 0.866i)15-s + (−0.5 − 0.866i)16-s + 0.999·18-s + (−0.866 + 1.5i)19-s + (−0.866 + 1.49i)20-s − 23-s + (−0.866 + 0.5i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.718736689\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.718736689\) |
\(L(1)\) |
|
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 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 1.73T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−8.629463124952444168815901111540, −8.148751356093869309253502945305, −7.32595216136246632854264020959, −6.35410251340992537824381783918, −6.16337252331402888312046417073, −5.31918430281186626348800863591, −4.27100168184256196346688372650, −3.39510869976281655838227019726, −2.38682055650684543456572838803, −1.66370152428346974851814850829,
1.49284217269098366485974919895, 2.36060904184755265515560302030, 2.77317902782277587216845953945, 3.94979833021120617646471566025, 4.72533628252201551206491413162, 5.41886377357936195570998954984, 6.19523619932347851100429118368, 6.97025898866061425636408602262, 8.284434855770687540672568023377, 9.008743372548478316186140320525