| L(s) = 1 | − 2-s − 4-s − 5-s + 7-s + 3·8-s + 10-s − 6·13-s − 14-s − 16-s − 2·17-s − 8·19-s + 20-s − 8·23-s + 25-s + 6·26-s − 28-s + 2·29-s + 4·31-s − 5·32-s + 2·34-s − 35-s − 2·37-s + 8·38-s − 3·40-s + 6·41-s + 4·43-s + 8·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 0.377·7-s + 1.06·8-s + 0.316·10-s − 1.66·13-s − 0.267·14-s − 1/4·16-s − 0.485·17-s − 1.83·19-s + 0.223·20-s − 1.66·23-s + 1/5·25-s + 1.17·26-s − 0.188·28-s + 0.371·29-s + 0.718·31-s − 0.883·32-s + 0.342·34-s − 0.169·35-s − 0.328·37-s + 1.29·38-s − 0.474·40-s + 0.937·41-s + 0.609·43-s + 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−10.96971784880108147181434505980, −10.15928312968820765258481052050, −9.326498061922569555987691090592, −8.261177977090796325600765542566, −7.73831918709431562684545235915, −6.48451110526696282246961711325, −4.87484917945516723546545158294, −4.17084275561163201389021211529, −2.17599002568151719086793905114, 0,
2.17599002568151719086793905114, 4.17084275561163201389021211529, 4.87484917945516723546545158294, 6.48451110526696282246961711325, 7.73831918709431562684545235915, 8.261177977090796325600765542566, 9.326498061922569555987691090592, 10.15928312968820765258481052050, 10.96971784880108147181434505980
