L(s) = 1 | − 3·5-s − 3·7-s − 11-s − 2·13-s + 5·17-s + 4·23-s + 4·25-s + 6·29-s + 2·31-s + 9·35-s + 8·37-s − 8·41-s − 13·43-s − 13·47-s + 2·49-s + 6·53-s + 3·55-s − 4·59-s + 13·61-s + 6·65-s + 4·67-s − 8·71-s − 3·73-s + 3·77-s + 4·79-s + 4·83-s − 15·85-s + ⋯ |
L(s) = 1 | − 1.34·5-s − 1.13·7-s − 0.301·11-s − 0.554·13-s + 1.21·17-s + 0.834·23-s + 4/5·25-s + 1.11·29-s + 0.359·31-s + 1.52·35-s + 1.31·37-s − 1.24·41-s − 1.98·43-s − 1.89·47-s + 2/7·49-s + 0.824·53-s + 0.404·55-s − 0.520·59-s + 1.66·61-s + 0.744·65-s + 0.488·67-s − 0.949·71-s − 0.351·73-s + 0.341·77-s + 0.450·79-s + 0.439·83-s − 1.62·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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
−13.18753826844008, −12.76968161821433, −12.28176269791012, −11.84478169090766, −11.52391154271865, −11.08059483028875, −10.21852164115725, −9.919285769450288, −9.828532350595789, −8.851723451135291, −8.512452968995712, −7.966926498861639, −7.625995421707879, −7.015366798878473, −6.628172181805148, −6.189085319316678, −5.376628505627907, −4.909739145502561, −4.501329609803112, −3.697746129265087, −3.272719867934446, −3.052463556847050, −2.292950702900367, −1.313784869420347, −0.6253002046622419, 0,
0.6253002046622419, 1.313784869420347, 2.292950702900367, 3.052463556847050, 3.272719867934446, 3.697746129265087, 4.501329609803112, 4.909739145502561, 5.376628505627907, 6.189085319316678, 6.628172181805148, 7.015366798878473, 7.625995421707879, 7.966926498861639, 8.512452968995712, 8.851723451135291, 9.828532350595789, 9.919285769450288, 10.21852164115725, 11.08059483028875, 11.52391154271865, 11.84478169090766, 12.28176269791012, 12.76968161821433, 13.18753826844008