| L(s) = 1 | − 3-s + 2·5-s − 2·7-s + 9-s + 4·13-s − 2·15-s − 2·17-s + 4·19-s + 2·21-s + 4·23-s − 25-s − 27-s + 6·29-s − 2·31-s − 4·35-s + 8·37-s − 4·39-s − 2·41-s + 4·43-s + 2·45-s + 12·47-s − 3·49-s + 2·51-s + 6·53-s − 4·57-s − 4·59-s − 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s + 1.10·13-s − 0.516·15-s − 0.485·17-s + 0.917·19-s + 0.436·21-s + 0.834·23-s − 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.359·31-s − 0.676·35-s + 1.31·37-s − 0.640·39-s − 0.312·41-s + 0.609·43-s + 0.298·45-s + 1.75·47-s − 3/7·49-s + 0.280·51-s + 0.824·53-s − 0.529·57-s − 0.520·59-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.453727716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.453727716\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−10.34935950175211712364487117087, −9.494482055881130926006645834294, −8.889403972374772412587674256600, −7.60982656432259660497620212594, −6.53729896264740462444493449031, −6.02952025279743756519610804548, −5.12505772656764585053702097555, −3.87120361857012739887619384713, −2.63713724931622987423343237649, −1.09527349773841298009626376824,
1.09527349773841298009626376824, 2.63713724931622987423343237649, 3.87120361857012739887619384713, 5.12505772656764585053702097555, 6.02952025279743756519610804548, 6.53729896264740462444493449031, 7.60982656432259660497620212594, 8.889403972374772412587674256600, 9.494482055881130926006645834294, 10.34935950175211712364487117087
