| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 4·11-s − 12-s − 2·13-s + 16-s + 2·17-s − 18-s − 4·19-s − 4·22-s + 8·23-s + 24-s + 2·26-s − 27-s − 2·29-s − 8·31-s − 32-s − 4·33-s − 2·34-s + 36-s − 37-s + 4·38-s + 2·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s − 0.554·13-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.852·22-s + 1.66·23-s + 0.204·24-s + 0.392·26-s − 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.176·32-s − 0.696·33-s − 0.342·34-s + 1/6·36-s − 0.164·37-s + 0.648·38-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 271950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 271950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5343124846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5343124846\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.79361984489095, −12.24482600966029, −11.63988932597715, −11.42950593158591, −10.99366017402346, −10.43243520831328, −9.992008114245762, −9.594947041117643, −9.013440876904138, −8.749821618331604, −8.177958727993941, −7.581280065873659, −6.999046555965275, −6.738627002914650, −6.405059069265290, −5.592599557489671, −5.221998565618456, −4.732565236985666, −4.000556792064310, −3.500054626722098, −2.995882513138411, −2.139918631960369, −1.576748461267224, −1.177740787401393, −0.2410533282319465,
0.2410533282319465, 1.177740787401393, 1.576748461267224, 2.139918631960369, 2.995882513138411, 3.500054626722098, 4.000556792064310, 4.732565236985666, 5.221998565618456, 5.592599557489671, 6.405059069265290, 6.738627002914650, 6.999046555965275, 7.581280065873659, 8.177958727993941, 8.749821618331604, 9.013440876904138, 9.594947041117643, 9.992008114245762, 10.43243520831328, 10.99366017402346, 11.42950593158591, 11.63988932597715, 12.24482600966029, 12.79361984489095
