L(s) = 1 | − 2.12·2-s + 3-s + 2.51·4-s + 2.60·5-s − 2.12·6-s − 0.251·7-s − 1.09·8-s + 9-s − 5.53·10-s − 2.53·11-s + 2.51·12-s + 1.09·13-s + 0.535·14-s + 2.60·15-s − 2.70·16-s + 17-s − 2.12·18-s − 3.26·19-s + 6.55·20-s − 0.251·21-s + 5.38·22-s + 5.88·23-s − 1.09·24-s + 1.79·25-s − 2.31·26-s + 27-s − 0.633·28-s + ⋯ |
L(s) = 1 | − 1.50·2-s + 0.577·3-s + 1.25·4-s + 1.16·5-s − 0.867·6-s − 0.0952·7-s − 0.386·8-s + 0.333·9-s − 1.75·10-s − 0.763·11-s + 0.726·12-s + 0.302·13-s + 0.143·14-s + 0.672·15-s − 0.676·16-s + 0.242·17-s − 0.500·18-s − 0.748·19-s + 1.46·20-s − 0.0549·21-s + 1.14·22-s + 1.22·23-s − 0.223·24-s + 0.358·25-s − 0.454·26-s + 0.192·27-s − 0.119·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
good | 2 | \( 1 + 2.12T + 2T^{2} \) |
| 5 | \( 1 - 2.60T + 5T^{2} \) |
| 7 | \( 1 + 0.251T + 7T^{2} \) |
| 11 | \( 1 + 2.53T + 11T^{2} \) |
| 13 | \( 1 - 1.09T + 13T^{2} \) |
| 19 | \( 1 + 3.26T + 19T^{2} \) |
| 23 | \( 1 - 5.88T + 23T^{2} \) |
| 29 | \( 1 + 1.51T + 29T^{2} \) |
| 31 | \( 1 + 4.09T + 31T^{2} \) |
| 37 | \( 1 + 0.827T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 8.35T + 43T^{2} \) |
| 47 | \( 1 + 2.43T + 47T^{2} \) |
| 53 | \( 1 + 5.60T + 53T^{2} \) |
| 59 | \( 1 - 8.39T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 + 16.5T + 71T^{2} \) |
| 73 | \( 1 - 2.12T + 73T^{2} \) |
| 79 | \( 1 + 3.58T + 79T^{2} \) |
| 83 | \( 1 - 0.369T + 83T^{2} \) |
| 89 | \( 1 + 0.659T + 89T^{2} \) |
| 97 | \( 1 - 7.05T + 97T^{2} \) |
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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
−7.70091084393467614818791417679, −6.90884983139663999700573912781, −6.49708384557964267296576768699, −5.45030907126181624754729879471, −4.86318755633003727541951937800, −3.61071889963971606744917041348, −2.71756189055260420526178363742, −1.94943389241970091723897128896, −1.36447020208268416960433615260, 0,
1.36447020208268416960433615260, 1.94943389241970091723897128896, 2.71756189055260420526178363742, 3.61071889963971606744917041348, 4.86318755633003727541951937800, 5.45030907126181624754729879471, 6.49708384557964267296576768699, 6.90884983139663999700573912781, 7.70091084393467614818791417679