| L(s) = 1 | + 3-s − 5-s − 4·7-s − 2·9-s + 11-s + 2·13-s − 15-s − 2·19-s − 4·21-s − 9·23-s − 4·25-s − 5·27-s − 4·29-s − 5·31-s + 33-s + 4·35-s + 9·37-s + 2·39-s + 2·41-s − 6·43-s + 2·45-s + 4·47-s + 9·49-s + 6·53-s − 55-s − 2·57-s − 5·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.51·7-s − 2/3·9-s + 0.301·11-s + 0.554·13-s − 0.258·15-s − 0.458·19-s − 0.872·21-s − 1.87·23-s − 4/5·25-s − 0.962·27-s − 0.742·29-s − 0.898·31-s + 0.174·33-s + 0.676·35-s + 1.47·37-s + 0.320·39-s + 0.312·41-s − 0.914·43-s + 0.298·45-s + 0.583·47-s + 9/7·49-s + 0.824·53-s − 0.134·55-s − 0.264·57-s − 0.650·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
| 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
−9.774943377255521456540277316793, −9.235033174044084876956156102006, −8.301962980665033555377946684791, −7.53313861270626628068378871368, −6.32653133042280981851578332829, −5.80229561888296893680026084263, −4.00866082421932220930913453450, −3.47503415787297074261780732064, −2.26341299200940895565096118841, 0,
2.26341299200940895565096118841, 3.47503415787297074261780732064, 4.00866082421932220930913453450, 5.80229561888296893680026084263, 6.32653133042280981851578332829, 7.53313861270626628068378871368, 8.301962980665033555377946684791, 9.235033174044084876956156102006, 9.774943377255521456540277316793
