| L(s) = 1 | − 3-s − 5-s + 9-s − 4·11-s + 2·13-s + 15-s + 2·17-s + 8·19-s + 25-s − 27-s − 2·29-s + 4·33-s − 6·37-s − 2·39-s − 2·41-s − 4·43-s − 45-s + 8·47-s − 2·51-s − 6·53-s + 4·55-s − 8·57-s + 2·61-s − 2·65-s − 4·67-s − 4·71-s + 2·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.258·15-s + 0.485·17-s + 1.83·19-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.696·33-s − 0.986·37-s − 0.320·39-s − 0.312·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s − 0.280·51-s − 0.824·53-s + 0.539·55-s − 1.05·57-s + 0.256·61-s − 0.248·65-s − 0.488·67-s − 0.474·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−15.70603265778085, −15.49651304881124, −14.68248766070672, −13.95838790390101, −13.61454614507606, −12.95047038945285, −12.41736585921309, −11.90913960684978, −11.33758511973150, −10.93520455546275, −10.18677849981698, −9.903435960220241, −9.089683933213653, −8.424248903850122, −7.820167394108376, −7.319377397934994, −6.846846212081952, −5.824667277367399, −5.515584634535640, −4.936686541579017, −4.184818942949480, −3.336138734006452, −2.929482332024614, −1.780435402646057, −0.9569285775959167, 0,
0.9569285775959167, 1.780435402646057, 2.929482332024614, 3.336138734006452, 4.184818942949480, 4.936686541579017, 5.515584634535640, 5.824667277367399, 6.846846212081952, 7.319377397934994, 7.820167394108376, 8.424248903850122, 9.089683933213653, 9.903435960220241, 10.18677849981698, 10.93520455546275, 11.33758511973150, 11.90913960684978, 12.41736585921309, 12.95047038945285, 13.61454614507606, 13.95838790390101, 14.68248766070672, 15.49651304881124, 15.70603265778085
