| L(s) = 1 | − 2·2-s + 2·4-s − 2·5-s + 4·10-s + 2·11-s − 13-s − 4·16-s − 19-s − 4·20-s − 4·22-s − 25-s + 2·26-s − 4·29-s − 9·31-s + 8·32-s + 3·37-s + 2·38-s − 10·41-s + 5·43-s + 4·44-s − 6·47-s + 2·50-s − 2·52-s − 12·53-s − 4·55-s + 8·58-s − 12·59-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.894·5-s + 1.26·10-s + 0.603·11-s − 0.277·13-s − 16-s − 0.229·19-s − 0.894·20-s − 0.852·22-s − 1/5·25-s + 0.392·26-s − 0.742·29-s − 1.61·31-s + 1.41·32-s + 0.493·37-s + 0.324·38-s − 1.56·41-s + 0.762·43-s + 0.603·44-s − 0.875·47-s + 0.282·50-s − 0.277·52-s − 1.64·53-s − 0.539·55-s + 1.05·58-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−10.62938272531856235232756893811, −9.532401741076686225901695719699, −8.970613305827843910423804944183, −7.920026489093004352764513852408, −7.42251681884249141719743547364, −6.33310252534575837626677109971, −4.73946994522750223509171420340, −3.54901712994340686137812719330, −1.75196450155362895800400842510, 0,
1.75196450155362895800400842510, 3.54901712994340686137812719330, 4.73946994522750223509171420340, 6.33310252534575837626677109971, 7.42251681884249141719743547364, 7.920026489093004352764513852408, 8.970613305827843910423804944183, 9.532401741076686225901695719699, 10.62938272531856235232756893811
