| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s − 3·11-s + 5·13-s + 14-s + 16-s − 3·17-s + 19-s + 20-s − 3·22-s − 9·23-s + 25-s + 5·26-s + 28-s − 3·29-s + 2·31-s + 32-s − 3·34-s + 35-s − 10·37-s + 38-s + 40-s + 3·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s − 0.904·11-s + 1.38·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.229·19-s + 0.223·20-s − 0.639·22-s − 1.87·23-s + 1/5·25-s + 0.980·26-s + 0.188·28-s − 0.557·29-s + 0.359·31-s + 0.176·32-s − 0.514·34-s + 0.169·35-s − 1.64·37-s + 0.162·38-s + 0.158·40-s + 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.894991292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.894991292\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−16.06896469479110, −15.79847825949536, −15.36849502511941, −14.56110959448566, −13.90009627336283, −13.63696056658032, −13.15409083767231, −12.41209181727019, −11.89051175970456, −11.20980251965896, −10.61635806100742, −10.28433921856721, −9.408571405024687, −8.610641935806534, −8.187047005845529, −7.415540257050977, −6.724610012812525, −5.963215346859981, −5.580770396501253, −4.891603283422776, −3.956577999075679, −3.600498640271619, −2.344798100036883, −2.032732336052138, −0.7992738747655472,
0.7992738747655472, 2.032732336052138, 2.344798100036883, 3.600498640271619, 3.956577999075679, 4.891603283422776, 5.580770396501253, 5.963215346859981, 6.724610012812525, 7.415540257050977, 8.187047005845529, 8.610641935806534, 9.408571405024687, 10.28433921856721, 10.61635806100742, 11.20980251965896, 11.89051175970456, 12.41209181727019, 13.15409083767231, 13.63696056658032, 13.90009627336283, 14.56110959448566, 15.36849502511941, 15.79847825949536, 16.06896469479110
