| L(s) = 1 | + 3-s − 2·5-s − 7-s + 9-s − 4·11-s − 6·13-s − 2·15-s − 2·17-s + 4·19-s − 21-s − 4·23-s − 25-s + 27-s − 2·29-s + 8·31-s − 4·33-s + 2·35-s − 10·37-s − 6·39-s − 2·41-s + 8·43-s − 2·45-s + 49-s − 2·51-s − 10·53-s + 8·55-s + 4·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s − 0.516·15-s − 0.485·17-s + 0.917·19-s − 0.218·21-s − 0.834·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.696·33-s + 0.338·35-s − 1.64·37-s − 0.960·39-s − 0.312·41-s + 1.21·43-s − 0.298·45-s + 1/7·49-s − 0.280·51-s − 1.37·53-s + 1.07·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.983394832839437657969162695512, −9.302688346205838211787606212991, −8.034900753827151172045193415181, −7.70276610727881458590413914623, −6.77473142578757723529467681346, −5.33934975205348273864801508199, −4.44304823928958265064368122586, −3.27099070744047974080065927216, −2.32898343675991772184414351496, 0,
2.32898343675991772184414351496, 3.27099070744047974080065927216, 4.44304823928958265064368122586, 5.33934975205348273864801508199, 6.77473142578757723529467681346, 7.70276610727881458590413914623, 8.034900753827151172045193415181, 9.302688346205838211787606212991, 9.983394832839437657969162695512
