| L(s) = 1 | − 2-s + 4-s − 8-s − 5·11-s + 13-s + 16-s + 8·19-s + 5·22-s − 4·23-s − 26-s − 5·29-s − 32-s − 2·37-s − 8·38-s − 9·41-s − 3·43-s − 5·44-s + 4·46-s − 8·47-s + 52-s + 5·53-s + 5·58-s − 13·59-s − 2·61-s + 64-s + 8·67-s − 2·71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.50·11-s + 0.277·13-s + 1/4·16-s + 1.83·19-s + 1.06·22-s − 0.834·23-s − 0.196·26-s − 0.928·29-s − 0.176·32-s − 0.328·37-s − 1.29·38-s − 1.40·41-s − 0.457·43-s − 0.753·44-s + 0.589·46-s − 1.16·47-s + 0.138·52-s + 0.686·53-s + 0.656·58-s − 1.69·59-s − 0.256·61-s + 1/8·64-s + 0.977·67-s − 0.237·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.27581054293816, −12.70815082580954, −12.05194319932896, −11.87603041511391, −11.18247713427817, −10.95262785477047, −10.28381160337557, −9.936485020560005, −9.673002968261836, −9.001769570300150, −8.559409697184477, −7.905453138588407, −7.823064949244077, −7.232962775930483, −6.777367513626132, −6.108088060199387, −5.608367734173053, −5.176998667582667, −4.801023108893486, −3.861654424481396, −3.403119381095095, −2.878696814756152, −2.360968532972515, −1.595376966941693, −1.210399683812319, 0, 0,
1.210399683812319, 1.595376966941693, 2.360968532972515, 2.878696814756152, 3.403119381095095, 3.861654424481396, 4.801023108893486, 5.176998667582667, 5.608367734173053, 6.108088060199387, 6.777367513626132, 7.232962775930483, 7.823064949244077, 7.905453138588407, 8.559409697184477, 9.001769570300150, 9.673002968261836, 9.936485020560005, 10.28381160337557, 10.95262785477047, 11.18247713427817, 11.87603041511391, 12.05194319932896, 12.70815082580954, 13.27581054293816
