| L(s) = 1 | − 1.69·2-s + 266.·3-s − 509.·4-s + 625·5-s − 452.·6-s − 2.40e3·7-s + 1.72e3·8-s + 5.15e4·9-s − 1.05e3·10-s + 6.52e4·11-s − 1.35e5·12-s + 2.98e4·13-s + 4.06e3·14-s + 1.66e5·15-s + 2.57e5·16-s − 1.39e5·17-s − 8.73e4·18-s − 4.87e5·19-s − 3.18e5·20-s − 6.40e5·21-s − 1.10e5·22-s + 2.19e6·23-s + 4.61e5·24-s + 3.90e5·25-s − 5.05e4·26-s + 8.51e6·27-s + 1.22e6·28-s + ⋯ |
| L(s) = 1 | − 0.0748·2-s + 1.90·3-s − 0.994·4-s + 0.447·5-s − 0.142·6-s − 0.377·7-s + 0.149·8-s + 2.62·9-s − 0.0334·10-s + 1.34·11-s − 1.89·12-s + 0.289·13-s + 0.0282·14-s + 0.850·15-s + 0.983·16-s − 0.405·17-s − 0.196·18-s − 0.857·19-s − 0.444·20-s − 0.719·21-s − 0.100·22-s + 1.63·23-s + 0.284·24-s + 0.200·25-s − 0.0216·26-s + 3.08·27-s + 0.375·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.201865302\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.201865302\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 625T \) |
| 7 | \( 1 + 2.40e3T \) |
| good | 2 | \( 1 + 1.69T + 512T^{2} \) |
| 3 | \( 1 - 266.T + 1.96e4T^{2} \) |
| 11 | \( 1 - 6.52e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 2.98e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.39e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 4.87e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.19e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.69e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.38e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.02e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.71e5T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.48e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.56e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.19e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 3.68e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.67e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.76e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.31e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.19e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.18e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.31e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.57e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 9.98e8T + 7.60e17T^{2} \) |
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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
−14.34763853636628521211917261611, −13.56091257359737410738407047342, −12.69917212985420750831239039157, −10.17493696947788865440449040613, −9.033409140870653540683834991799, −8.647016079525162374323988209369, −6.87434317715422781686466275969, −4.38225455168110665422347289810, −3.18314066931313778056885995402, −1.42088477882847551476492346515,
1.42088477882847551476492346515, 3.18314066931313778056885995402, 4.38225455168110665422347289810, 6.87434317715422781686466275969, 8.647016079525162374323988209369, 9.033409140870653540683834991799, 10.17493696947788865440449040613, 12.69917212985420750831239039157, 13.56091257359737410738407047342, 14.34763853636628521211917261611
