| L(s) = 1 | − 3-s − 3·5-s − 3·7-s + 9-s + 11-s − 13-s + 3·15-s − 4·17-s − 6·19-s + 3·21-s − 7·23-s + 4·25-s − 27-s + 3·29-s + 4·31-s − 33-s + 9·35-s − 2·37-s + 39-s − 9·41-s − 7·43-s − 3·45-s − 4·47-s + 2·49-s + 4·51-s − 10·53-s − 3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 1.13·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s + 0.774·15-s − 0.970·17-s − 1.37·19-s + 0.654·21-s − 1.45·23-s + 4/5·25-s − 0.192·27-s + 0.557·29-s + 0.718·31-s − 0.174·33-s + 1.52·35-s − 0.328·37-s + 0.160·39-s − 1.40·41-s − 1.06·43-s − 0.447·45-s − 0.583·47-s + 2/7·49-s + 0.560·51-s − 1.37·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.11596382785954246848820932143, −6.44699349970762983436422058576, −6.23230756472595548724225650893, −4.92538406646070237172444538653, −4.32771019804691496157615304291, −3.72168130576801886543372118781, −2.91054577632091133387539832616, −1.74119798283467300507242097280, 0, 0,
1.74119798283467300507242097280, 2.91054577632091133387539832616, 3.72168130576801886543372118781, 4.32771019804691496157615304291, 4.92538406646070237172444538653, 6.23230756472595548724225650893, 6.44699349970762983436422058576, 7.11596382785954246848820932143
