| L(s) = 1 | − 9·5-s + 31·7-s + 15·11-s − 37·13-s − 42·17-s + 28·19-s − 195·23-s − 44·25-s + 111·29-s + 205·31-s − 279·35-s − 166·37-s − 261·41-s + 43·43-s − 177·47-s + 618·49-s + 114·53-s − 135·55-s − 159·59-s + 191·61-s + 333·65-s + 421·67-s − 156·71-s + 182·73-s + 465·77-s − 1.13e3·79-s + 1.08e3·83-s + ⋯ |
| L(s) = 1 | − 0.804·5-s + 1.67·7-s + 0.411·11-s − 0.789·13-s − 0.599·17-s + 0.338·19-s − 1.76·23-s − 0.351·25-s + 0.710·29-s + 1.18·31-s − 1.34·35-s − 0.737·37-s − 0.994·41-s + 0.152·43-s − 0.549·47-s + 1.80·49-s + 0.295·53-s − 0.330·55-s − 0.350·59-s + 0.400·61-s + 0.635·65-s + 0.767·67-s − 0.260·71-s + 0.291·73-s + 0.688·77-s − 1.61·79-s + 1.43·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 9 T + p^{3} T^{2} \) |
| 7 | \( 1 - 31 T + p^{3} T^{2} \) |
| 11 | \( 1 - 15 T + p^{3} T^{2} \) |
| 13 | \( 1 + 37 T + p^{3} T^{2} \) |
| 17 | \( 1 + 42 T + p^{3} T^{2} \) |
| 19 | \( 1 - 28 T + p^{3} T^{2} \) |
| 23 | \( 1 + 195 T + p^{3} T^{2} \) |
| 29 | \( 1 - 111 T + p^{3} T^{2} \) |
| 31 | \( 1 - 205 T + p^{3} T^{2} \) |
| 37 | \( 1 + 166 T + p^{3} T^{2} \) |
| 41 | \( 1 + 261 T + p^{3} T^{2} \) |
| 43 | \( 1 - p T + p^{3} T^{2} \) |
| 47 | \( 1 + 177 T + p^{3} T^{2} \) |
| 53 | \( 1 - 114 T + p^{3} T^{2} \) |
| 59 | \( 1 + 159 T + p^{3} T^{2} \) |
| 61 | \( 1 - 191 T + p^{3} T^{2} \) |
| 67 | \( 1 - 421 T + p^{3} T^{2} \) |
| 71 | \( 1 + 156 T + p^{3} T^{2} \) |
| 73 | \( 1 - 182 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1133 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1083 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1050 T + p^{3} T^{2} \) |
| 97 | \( 1 + 901 T + p^{3} T^{2} \) |
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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
−8.555952263065632215573131097331, −8.120433312263270411665501158933, −7.44834574192343762624598795112, −6.49043535108031052114607135032, −5.29314082416548591816985004384, −4.54509982270863392689303606111, −3.86939324004544099989531941640, −2.41890159076111044036778997660, −1.42106134641550147578522896322, 0,
1.42106134641550147578522896322, 2.41890159076111044036778997660, 3.86939324004544099989531941640, 4.54509982270863392689303606111, 5.29314082416548591816985004384, 6.49043535108031052114607135032, 7.44834574192343762624598795112, 8.120433312263270411665501158933, 8.555952263065632215573131097331
