L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s − 2·11-s + 2·13-s + 14-s + 16-s − 2·17-s − 4·19-s + 20-s − 2·22-s + 23-s + 25-s + 2·26-s + 28-s − 8·29-s + 2·31-s + 32-s − 2·34-s + 35-s + 4·37-s − 4·38-s + 40-s − 2·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s − 0.603·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.917·19-s + 0.223·20-s − 0.426·22-s + 0.208·23-s + 1/5·25-s + 0.392·26-s + 0.188·28-s − 1.48·29-s + 0.359·31-s + 0.176·32-s − 0.342·34-s + 0.169·35-s + 0.657·37-s − 0.648·38-s + 0.158·40-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.41594889957226, −15.69175374428488, −15.07652711502683, −14.82795210345665, −14.10451378435958, −13.49676912808974, −13.01978107850617, −12.78173235970930, −11.83291550220399, −11.27741980645363, −10.88780653429985, −10.22646172964067, −9.614065126040238, −8.826484572507025, −8.262711497202245, −7.631857449395352, −6.891679456260037, −6.217056116618116, −5.791477193776060, −4.900591614623954, −4.564047660516994, −3.620556640324268, −2.957229462035392, −2.055440033532021, −1.496290780444935, 0,
1.496290780444935, 2.055440033532021, 2.957229462035392, 3.620556640324268, 4.564047660516994, 4.900591614623954, 5.791477193776060, 6.217056116618116, 6.891679456260037, 7.631857449395352, 8.262711497202245, 8.826484572507025, 9.614065126040238, 10.22646172964067, 10.88780653429985, 11.27741980645363, 11.83291550220399, 12.78173235970930, 13.01978107850617, 13.49676912808974, 14.10451378435958, 14.82795210345665, 15.07652711502683, 15.69175374428488, 16.41594889957226