L(s) = 1 | + 2-s + 3-s + 4-s + 3.56·5-s + 6-s − 2.46·7-s + 8-s + 9-s + 3.56·10-s + 2.23·11-s + 12-s + 6.98·13-s − 2.46·14-s + 3.56·15-s + 16-s − 17-s + 18-s − 3.26·19-s + 3.56·20-s − 2.46·21-s + 2.23·22-s − 0.970·23-s + 24-s + 7.70·25-s + 6.98·26-s + 27-s − 2.46·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.59·5-s + 0.408·6-s − 0.932·7-s + 0.353·8-s + 0.333·9-s + 1.12·10-s + 0.674·11-s + 0.288·12-s + 1.93·13-s − 0.659·14-s + 0.920·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s − 0.748·19-s + 0.796·20-s − 0.538·21-s + 0.476·22-s − 0.202·23-s + 0.204·24-s + 1.54·25-s + 1.37·26-s + 0.192·27-s − 0.466·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.739343797\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.739343797\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
good | 5 | \( 1 - 3.56T + 5T^{2} \) |
| 7 | \( 1 + 2.46T + 7T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 - 6.98T + 13T^{2} \) |
| 19 | \( 1 + 3.26T + 19T^{2} \) |
| 23 | \( 1 + 0.970T + 23T^{2} \) |
| 29 | \( 1 - 5.56T + 29T^{2} \) |
| 31 | \( 1 - 2.91T + 31T^{2} \) |
| 37 | \( 1 + 6.24T + 37T^{2} \) |
| 41 | \( 1 + 5.50T + 41T^{2} \) |
| 43 | \( 1 + 6.05T + 43T^{2} \) |
| 47 | \( 1 - 0.503T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 61 | \( 1 - 6.17T + 61T^{2} \) |
| 67 | \( 1 - 2.12T + 67T^{2} \) |
| 71 | \( 1 + 3.41T + 71T^{2} \) |
| 73 | \( 1 + 1.74T + 73T^{2} \) |
| 79 | \( 1 + 0.425T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 - 7.98T + 97T^{2} \) |
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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
−8.408554025873444965394421228322, −6.83948671926808120626269189170, −6.60691559140846477140769313806, −6.03420450408170628172889541298, −5.37197432011141599675584458514, −4.29281395093519668684264845353, −3.57914468251365826980290066488, −2.87951464201862583474007186721, −1.96659329965428230990828543494, −1.23125530890269007388235354459,
1.23125530890269007388235354459, 1.96659329965428230990828543494, 2.87951464201862583474007186721, 3.57914468251365826980290066488, 4.29281395093519668684264845353, 5.37197432011141599675584458514, 6.03420450408170628172889541298, 6.60691559140846477140769313806, 6.83948671926808120626269189170, 8.408554025873444965394421228322