| L(s) = 1 | − 2.34·2-s + 3.28·3-s + 3.50·4-s + 3.73·5-s − 7.71·6-s + 3.13·7-s − 3.53·8-s + 7.79·9-s − 8.77·10-s + 11-s + 11.5·12-s + 0.157·13-s − 7.35·14-s + 12.2·15-s + 1.28·16-s + 17-s − 18.3·18-s − 5.83·19-s + 13.1·20-s + 10.3·21-s − 2.34·22-s + 0.802·23-s − 11.6·24-s + 8.98·25-s − 0.369·26-s + 15.7·27-s + 10.9·28-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 1.89·3-s + 1.75·4-s + 1.67·5-s − 3.14·6-s + 1.18·7-s − 1.24·8-s + 2.59·9-s − 2.77·10-s + 0.301·11-s + 3.32·12-s + 0.0436·13-s − 1.96·14-s + 3.17·15-s + 0.320·16-s + 0.242·17-s − 4.31·18-s − 1.33·19-s + 2.93·20-s + 2.24·21-s − 0.500·22-s + 0.167·23-s − 2.37·24-s + 1.79·25-s − 0.0724·26-s + 3.03·27-s + 2.07·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.550414689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.550414689\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 2.34T + 2T^{2} \) |
| 3 | \( 1 - 3.28T + 3T^{2} \) |
| 5 | \( 1 - 3.73T + 5T^{2} \) |
| 7 | \( 1 - 3.13T + 7T^{2} \) |
| 13 | \( 1 - 0.157T + 13T^{2} \) |
| 19 | \( 1 + 5.83T + 19T^{2} \) |
| 23 | \( 1 - 0.802T + 23T^{2} \) |
| 29 | \( 1 - 1.68T + 29T^{2} \) |
| 31 | \( 1 - 0.923T + 31T^{2} \) |
| 37 | \( 1 - 7.42T + 37T^{2} \) |
| 41 | \( 1 + 5.49T + 41T^{2} \) |
| 47 | \( 1 + 6.58T + 47T^{2} \) |
| 53 | \( 1 + 4.16T + 53T^{2} \) |
| 59 | \( 1 + 0.00333T + 59T^{2} \) |
| 61 | \( 1 - 7.96T + 61T^{2} \) |
| 67 | \( 1 + 2.69T + 67T^{2} \) |
| 71 | \( 1 + 0.778T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 + 7.37T + 79T^{2} \) |
| 83 | \( 1 - 6.85T + 83T^{2} \) |
| 89 | \( 1 + 9.62T + 89T^{2} \) |
| 97 | \( 1 - 5.96T + 97T^{2} \) |
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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
−8.215028977206536624337043453497, −7.49952620186974967704031180317, −6.78402307655172693653886423880, −6.16395918489297097223055807309, −4.94345086299236171321799423489, −4.18547598012274198717090418884, −2.88877813168523752347315748993, −2.31995333427357147683455362429, −1.66893526541161428302077675760, −1.29065314398540561233596851331,
1.29065314398540561233596851331, 1.66893526541161428302077675760, 2.31995333427357147683455362429, 2.88877813168523752347315748993, 4.18547598012274198717090418884, 4.94345086299236171321799423489, 6.16395918489297097223055807309, 6.78402307655172693653886423880, 7.49952620186974967704031180317, 8.215028977206536624337043453497
