| L(s) = 1 | − 2·3-s − 2·5-s + 7-s + 9-s − 2·13-s + 4·15-s − 2·17-s − 4·19-s − 2·21-s − 25-s + 4·27-s − 2·29-s − 10·31-s − 2·35-s + 8·37-s + 4·39-s + 2·41-s − 4·43-s − 2·45-s − 6·47-s + 49-s + 4·51-s − 12·53-s + 8·57-s + 10·59-s − 6·61-s + 63-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.554·13-s + 1.03·15-s − 0.485·17-s − 0.917·19-s − 0.436·21-s − 1/5·25-s + 0.769·27-s − 0.371·29-s − 1.79·31-s − 0.338·35-s + 1.31·37-s + 0.640·39-s + 0.312·41-s − 0.609·43-s − 0.298·45-s − 0.875·47-s + 1/7·49-s + 0.560·51-s − 1.64·53-s + 1.05·57-s + 1.30·59-s − 0.768·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−15.84393077805269, −15.08330490030524, −14.71723907091218, −14.34121772371248, −13.33193534727538, −12.91443483280786, −12.39144713684136, −11.83901223163971, −11.36656996401016, −11.03730573119840, −10.59760050187626, −9.811196116341896, −9.230888471252907, −8.494085730024725, −8.011768594041706, −7.347864893994395, −6.899091638543511, −6.120263524175794, −5.713197045437302, −4.927254281651345, −4.488308728676421, −3.906530796281422, −3.060525423229779, −2.167761765442634, −1.326210487864211, 0, 0,
1.326210487864211, 2.167761765442634, 3.060525423229779, 3.906530796281422, 4.488308728676421, 4.927254281651345, 5.713197045437302, 6.120263524175794, 6.899091638543511, 7.347864893994395, 8.011768594041706, 8.494085730024725, 9.230888471252907, 9.811196116341896, 10.59760050187626, 11.03730573119840, 11.36656996401016, 11.83901223163971, 12.39144713684136, 12.91443483280786, 13.33193534727538, 14.34121772371248, 14.71723907091218, 15.08330490030524, 15.84393077805269
