| L(s) = 1 | + 3-s + 2·5-s + 9-s − 4·11-s − 2·13-s + 2·15-s + 6·17-s − 4·19-s − 25-s + 27-s − 2·29-s − 4·31-s − 4·33-s + 2·37-s − 2·39-s − 2·41-s − 4·43-s + 2·45-s − 8·47-s + 6·51-s − 10·53-s − 8·55-s − 4·57-s − 4·59-s + 6·61-s − 4·65-s − 4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.516·15-s + 1.45·17-s − 0.917·19-s − 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.696·33-s + 0.328·37-s − 0.320·39-s − 0.312·41-s − 0.609·43-s + 0.298·45-s − 1.16·47-s + 0.840·51-s − 1.37·53-s − 1.07·55-s − 0.529·57-s − 0.520·59-s + 0.768·61-s − 0.496·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−7.67162363983023832897628035967, −6.65666333493910668914018217630, −5.95132791728059396352303876529, −5.28771285154964854479758637046, −4.73162153489596639015077930833, −3.65210147795777652202738378443, −2.94780004685597121841620199035, −2.19759371576614044642747592137, −1.50661705836964438501076917783, 0,
1.50661705836964438501076917783, 2.19759371576614044642747592137, 2.94780004685597121841620199035, 3.65210147795777652202738378443, 4.73162153489596639015077930833, 5.28771285154964854479758637046, 5.95132791728059396352303876529, 6.65666333493910668914018217630, 7.67162363983023832897628035967
