L(s) = 1 | + 2-s − 3-s + 4-s − 1.37·5-s − 6-s − 2.65·7-s + 8-s − 2·9-s − 1.37·10-s + 5.92·11-s − 12-s − 2.54·13-s − 2.65·14-s + 1.37·15-s + 16-s + 3.40·17-s − 2·18-s + 2.37·19-s − 1.37·20-s + 2.65·21-s + 5.92·22-s − 0.348·23-s − 24-s − 3.10·25-s − 2.54·26-s + 5·27-s − 2.65·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.615·5-s − 0.408·6-s − 1.00·7-s + 0.353·8-s − 0.666·9-s − 0.435·10-s + 1.78·11-s − 0.288·12-s − 0.706·13-s − 0.708·14-s + 0.355·15-s + 0.250·16-s + 0.825·17-s − 0.471·18-s + 0.545·19-s − 0.307·20-s + 0.578·21-s + 1.26·22-s − 0.0727·23-s − 0.204·24-s − 0.620·25-s − 0.499·26-s + 0.962·27-s − 0.501·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2011 | \( 1 - T \) |
good | 3 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 + 1.37T + 5T^{2} \) |
| 7 | \( 1 + 2.65T + 7T^{2} \) |
| 11 | \( 1 - 5.92T + 11T^{2} \) |
| 13 | \( 1 + 2.54T + 13T^{2} \) |
| 17 | \( 1 - 3.40T + 17T^{2} \) |
| 19 | \( 1 - 2.37T + 19T^{2} \) |
| 23 | \( 1 + 0.348T + 23T^{2} \) |
| 29 | \( 1 + 4.34T + 29T^{2} \) |
| 31 | \( 1 - 8.70T + 31T^{2} \) |
| 37 | \( 1 + 5.92T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 + 4.27T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 4.92T + 53T^{2} \) |
| 59 | \( 1 + 4.03T + 59T^{2} \) |
| 61 | \( 1 - 1.82T + 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 + 1.45T + 71T^{2} \) |
| 73 | \( 1 + 6.58T + 73T^{2} \) |
| 79 | \( 1 + 3.25T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 + 3.70T + 97T^{2} \) |
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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
−7.971062635793831303586782743765, −6.96777032702007001899118250526, −6.62309208920454445400055414969, −5.81486099072706034374290631077, −5.17868827713042084331422082530, −4.14866734392034142677055079622, −3.53401012009422937020358616168, −2.82462464203504724982360661771, −1.36402112361311783527842405851, 0,
1.36402112361311783527842405851, 2.82462464203504724982360661771, 3.53401012009422937020358616168, 4.14866734392034142677055079622, 5.17868827713042084331422082530, 5.81486099072706034374290631077, 6.62309208920454445400055414969, 6.96777032702007001899118250526, 7.971062635793831303586782743765