| L(s) = 1 | + 3-s + 7-s + 9-s − 4·11-s − 6·13-s − 2·19-s + 21-s + 23-s − 5·25-s + 27-s + 29-s − 8·31-s − 4·33-s − 9·37-s − 6·39-s + 6·41-s + 11·43-s − 12·47-s + 49-s − 53-s − 2·57-s − 6·59-s − 6·61-s + 63-s − 4·67-s + 69-s − 5·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s − 0.458·19-s + 0.218·21-s + 0.208·23-s − 25-s + 0.192·27-s + 0.185·29-s − 1.43·31-s − 0.696·33-s − 1.47·37-s − 0.960·39-s + 0.937·41-s + 1.67·43-s − 1.75·47-s + 1/7·49-s − 0.137·53-s − 0.264·57-s − 0.781·59-s − 0.768·61-s + 0.125·63-s − 0.488·67-s + 0.120·69-s − 0.593·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−12.72645657886077, −12.27676223663400, −11.91844473597408, −11.25431379887248, −10.83673993759807, −10.31668054660377, −10.09245153117964, −9.455055358289238, −9.061425529539110, −8.679866174646555, −7.963919721057682, −7.546785845865960, −7.503609898930617, −6.908880835301029, −6.106166552286422, −5.723959821588511, −5.131329323061205, −4.643314135390228, −4.423939285421214, −3.506985374944271, −3.158483973364292, −2.522531539399435, −1.984818054378543, −1.763776302164619, −0.5962129547010953, 0,
0.5962129547010953, 1.763776302164619, 1.984818054378543, 2.522531539399435, 3.158483973364292, 3.506985374944271, 4.423939285421214, 4.643314135390228, 5.131329323061205, 5.723959821588511, 6.106166552286422, 6.908880835301029, 7.503609898930617, 7.546785845865960, 7.963919721057682, 8.679866174646555, 9.061425529539110, 9.455055358289238, 10.09245153117964, 10.31668054660377, 10.83673993759807, 11.25431379887248, 11.91844473597408, 12.27676223663400, 12.72645657886077
