| L(s) = 1 | − 2-s − 3.15·3-s + 4-s − 2.25·5-s + 3.15·6-s − 1.49·7-s − 8-s + 6.93·9-s + 2.25·10-s − 3.03·11-s − 3.15·12-s + 3.87·13-s + 1.49·14-s + 7.12·15-s + 16-s − 3.17·17-s − 6.93·18-s − 19-s − 2.25·20-s + 4.72·21-s + 3.03·22-s + 3.24·23-s + 3.15·24-s + 0.104·25-s − 3.87·26-s − 12.4·27-s − 1.49·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.82·3-s + 0.5·4-s − 1.01·5-s + 1.28·6-s − 0.566·7-s − 0.353·8-s + 2.31·9-s + 0.714·10-s − 0.915·11-s − 0.910·12-s + 1.07·13-s + 0.400·14-s + 1.83·15-s + 0.250·16-s − 0.770·17-s − 1.63·18-s − 0.229·19-s − 0.505·20-s + 1.03·21-s + 0.647·22-s + 0.677·23-s + 0.643·24-s + 0.0208·25-s − 0.760·26-s − 2.39·27-s − 0.283·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 3.15T + 3T^{2} \) |
| 5 | \( 1 + 2.25T + 5T^{2} \) |
| 7 | \( 1 + 1.49T + 7T^{2} \) |
| 11 | \( 1 + 3.03T + 11T^{2} \) |
| 13 | \( 1 - 3.87T + 13T^{2} \) |
| 17 | \( 1 + 3.17T + 17T^{2} \) |
| 23 | \( 1 - 3.24T + 23T^{2} \) |
| 29 | \( 1 - 1.46T + 29T^{2} \) |
| 31 | \( 1 + 10.7T + 31T^{2} \) |
| 37 | \( 1 - 3.23T + 37T^{2} \) |
| 41 | \( 1 - 3.81T + 41T^{2} \) |
| 43 | \( 1 - 1.14T + 43T^{2} \) |
| 47 | \( 1 + 0.559T + 47T^{2} \) |
| 53 | \( 1 + 1.40T + 53T^{2} \) |
| 59 | \( 1 + 0.460T + 59T^{2} \) |
| 61 | \( 1 + 2.40T + 61T^{2} \) |
| 67 | \( 1 + 9.34T + 67T^{2} \) |
| 71 | \( 1 - 4.71T + 71T^{2} \) |
| 73 | \( 1 + 8.63T + 73T^{2} \) |
| 79 | \( 1 - 6.74T + 79T^{2} \) |
| 83 | \( 1 - 2.80T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 - 7.51T + 97T^{2} \) |
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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
−7.44806494108182201706306296577, −6.75987960322119459167175225813, −6.18450442654322462846088293658, −5.58700579556002372845747855812, −4.76765172450369432455438998046, −4.02828203079071836014154630039, −3.20067008122439609623417446637, −1.85918416288452683196556759977, −0.71541135313625066094150948922, 0,
0.71541135313625066094150948922, 1.85918416288452683196556759977, 3.20067008122439609623417446637, 4.02828203079071836014154630039, 4.76765172450369432455438998046, 5.58700579556002372845747855812, 6.18450442654322462846088293658, 6.75987960322119459167175225813, 7.44806494108182201706306296577
