| L(s) = 1 | + 3-s − 3·5-s − 7-s + 9-s + 11-s + 13-s − 3·15-s − 8·17-s + 6·19-s − 21-s + 23-s + 4·25-s + 27-s + 29-s + 33-s + 3·35-s + 6·37-s + 39-s − 11·41-s + 11·43-s − 3·45-s + 12·47-s − 6·49-s − 8·51-s + 2·53-s − 3·55-s + 6·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s − 0.774·15-s − 1.94·17-s + 1.37·19-s − 0.218·21-s + 0.208·23-s + 4/5·25-s + 0.192·27-s + 0.185·29-s + 0.174·33-s + 0.507·35-s + 0.986·37-s + 0.160·39-s − 1.71·41-s + 1.67·43-s − 0.447·45-s + 1.75·47-s − 6/7·49-s − 1.12·51-s + 0.274·53-s − 0.404·55-s + 0.794·57-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 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−7.58721884149519134371069115292, −7.08216591678405652133168635482, −6.43632615459467721294847347956, −5.43710125712846921251650263852, −4.37047074122306217413915797122, −4.07708271632966228265522609931, −3.19153085318953467672435610938, −2.51372013192647078165387320341, −1.21431222331136816130314828535, 0,
1.21431222331136816130314828535, 2.51372013192647078165387320341, 3.19153085318953467672435610938, 4.07708271632966228265522609931, 4.37047074122306217413915797122, 5.43710125712846921251650263852, 6.43632615459467721294847347956, 7.08216591678405652133168635482, 7.58721884149519134371069115292
