L(s) = 1 | − 2-s − 4-s − 5-s − 2·7-s + 3·8-s + 10-s + 13-s + 2·14-s − 16-s + 5·17-s + 6·19-s + 20-s − 2·23-s − 4·25-s − 26-s + 2·28-s − 9·29-s − 2·31-s − 5·32-s − 5·34-s + 2·35-s − 3·37-s − 6·38-s − 3·40-s + 5·41-s + 2·46-s − 2·47-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s − 0.755·7-s + 1.06·8-s + 0.316·10-s + 0.277·13-s + 0.534·14-s − 1/4·16-s + 1.21·17-s + 1.37·19-s + 0.223·20-s − 0.417·23-s − 4/5·25-s − 0.196·26-s + 0.377·28-s − 1.67·29-s − 0.359·31-s − 0.883·32-s − 0.857·34-s + 0.338·35-s − 0.493·37-s − 0.973·38-s − 0.474·40-s + 0.780·41-s + 0.294·46-s − 0.291·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−9.633324136261317362095613964778, −8.731578058982900025567463262875, −7.69035551562908784013462192082, −7.43742239849917419501134588307, −6.02373904066748853888838701833, −5.21092257719357715807199072115, −3.96049656444482134108637352404, −3.26020657517957887232980598752, −1.45371564777531463821755560879, 0,
1.45371564777531463821755560879, 3.26020657517957887232980598752, 3.96049656444482134108637352404, 5.21092257719357715807199072115, 6.02373904066748853888838701833, 7.43742239849917419501134588307, 7.69035551562908784013462192082, 8.731578058982900025567463262875, 9.633324136261317362095613964778