| L(s) = 1 | − 2·2-s + 2·4-s + 4·5-s + 7-s − 8·10-s − 2·13-s − 2·14-s − 4·16-s − 5·17-s + 5·19-s + 8·20-s + 11·25-s + 4·26-s + 2·28-s − 29-s − 3·31-s + 8·32-s + 10·34-s + 4·35-s − 8·37-s − 10·38-s + 10·41-s − 4·43-s + 8·47-s + 49-s − 22·50-s − 4·52-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 1.78·5-s + 0.377·7-s − 2.52·10-s − 0.554·13-s − 0.534·14-s − 16-s − 1.21·17-s + 1.14·19-s + 1.78·20-s + 11/5·25-s + 0.784·26-s + 0.377·28-s − 0.185·29-s − 0.538·31-s + 1.41·32-s + 1.71·34-s + 0.676·35-s − 1.31·37-s − 1.62·38-s + 1.56·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s − 3.11·50-s − 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.656125361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.656125361\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−13.13241786493711, −12.42847873611449, −11.97908843145085, −11.27095791636905, −10.91582378974465, −10.48092221265804, −10.04962107988087, −9.628039854754199, −9.246800713172216, −8.829739428053616, −8.569865172937410, −7.755300903181731, −7.219185403450700, −7.033688159752555, −6.344658681449318, −5.761266114556169, −5.412428962391457, −4.729384763875793, −4.358137350635683, −3.344151896901376, −2.614300007179398, −2.189625851090071, −1.716737582048357, −1.201707566356128, −0.4638896077784511,
0.4638896077784511, 1.201707566356128, 1.716737582048357, 2.189625851090071, 2.614300007179398, 3.344151896901376, 4.358137350635683, 4.729384763875793, 5.412428962391457, 5.761266114556169, 6.344658681449318, 7.033688159752555, 7.219185403450700, 7.755300903181731, 8.569865172937410, 8.829739428053616, 9.246800713172216, 9.628039854754199, 10.04962107988087, 10.48092221265804, 10.91582378974465, 11.27095791636905, 11.97908843145085, 12.42847873611449, 13.13241786493711
