L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s − 3·9-s + 2·10-s + 2·13-s + 16-s − 3·18-s − 4·19-s + 2·20-s − 25-s + 2·26-s + 6·29-s + 32-s − 3·36-s + 6·37-s − 4·38-s + 2·40-s − 6·41-s − 12·43-s − 6·45-s − 8·47-s − 50-s + 2·52-s − 2·53-s + 6·58-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s − 9-s + 0.632·10-s + 0.554·13-s + 1/4·16-s − 0.707·18-s − 0.917·19-s + 0.447·20-s − 1/5·25-s + 0.392·26-s + 1.11·29-s + 0.176·32-s − 1/2·36-s + 0.986·37-s − 0.648·38-s + 0.316·40-s − 0.937·41-s − 1.82·43-s − 0.894·45-s − 1.16·47-s − 0.141·50-s + 0.277·52-s − 0.274·53-s + 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.34220483437642, −14.80876264289998, −14.37005070231958, −13.81317129100496, −13.43310946579989, −12.99986322270227, −12.30291167402746, −11.79226512460799, −11.16329603366486, −10.83952667827270, −9.972501520803618, −9.744258686034757, −8.839346559880173, −8.293807603166725, −7.967445317669717, −6.685763271351715, −6.605673582410295, −5.926695384849067, −5.371752605210680, −4.834561939190213, −4.059448972057715, −3.311977933432902, −2.708090491492663, −2.030471001545682, −1.294387649284134, 0,
1.294387649284134, 2.030471001545682, 2.708090491492663, 3.311977933432902, 4.059448972057715, 4.834561939190213, 5.371752605210680, 5.926695384849067, 6.605673582410295, 6.685763271351715, 7.967445317669717, 8.293807603166725, 8.839346559880173, 9.744258686034757, 9.972501520803618, 10.83952667827270, 11.16329603366486, 11.79226512460799, 12.30291167402746, 12.99986322270227, 13.43310946579989, 13.81317129100496, 14.37005070231958, 14.80876264289998, 15.34220483437642