| L(s) = 1 | − 0.556·2-s − 0.855·3-s − 1.69·4-s − 5-s + 0.476·6-s + 7-s + 2.05·8-s − 2.26·9-s + 0.556·10-s + 3.86·11-s + 1.44·12-s + 6.16·13-s − 0.556·14-s + 0.855·15-s + 2.23·16-s − 0.562·17-s + 1.26·18-s − 7.14·19-s + 1.69·20-s − 0.855·21-s − 2.15·22-s − 7.00·23-s − 1.75·24-s + 25-s − 3.42·26-s + 4.50·27-s − 1.69·28-s + ⋯ |
| L(s) = 1 | − 0.393·2-s − 0.493·3-s − 0.845·4-s − 0.447·5-s + 0.194·6-s + 0.377·7-s + 0.726·8-s − 0.756·9-s + 0.176·10-s + 1.16·11-s + 0.417·12-s + 1.70·13-s − 0.148·14-s + 0.220·15-s + 0.559·16-s − 0.136·17-s + 0.297·18-s − 1.64·19-s + 0.377·20-s − 0.186·21-s − 0.458·22-s − 1.46·23-s − 0.358·24-s + 0.200·25-s − 0.672·26-s + 0.867·27-s − 0.319·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 + 0.556T + 2T^{2} \) |
| 3 | \( 1 + 0.855T + 3T^{2} \) |
| 11 | \( 1 - 3.86T + 11T^{2} \) |
| 13 | \( 1 - 6.16T + 13T^{2} \) |
| 17 | \( 1 + 0.562T + 17T^{2} \) |
| 19 | \( 1 + 7.14T + 19T^{2} \) |
| 23 | \( 1 + 7.00T + 23T^{2} \) |
| 29 | \( 1 + 0.249T + 29T^{2} \) |
| 31 | \( 1 - 2.99T + 31T^{2} \) |
| 37 | \( 1 + 4.66T + 37T^{2} \) |
| 41 | \( 1 + 5.67T + 41T^{2} \) |
| 43 | \( 1 - 8.89T + 43T^{2} \) |
| 47 | \( 1 - 3.83T + 47T^{2} \) |
| 53 | \( 1 - 4.47T + 53T^{2} \) |
| 59 | \( 1 + 4.21T + 59T^{2} \) |
| 61 | \( 1 + 7.41T + 61T^{2} \) |
| 67 | \( 1 + 1.37T + 67T^{2} \) |
| 71 | \( 1 - 1.22T + 71T^{2} \) |
| 73 | \( 1 + 2.45T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 4.50T + 83T^{2} \) |
| 89 | \( 1 - 9.13T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−7.73253269809473547449052688065, −6.63173300142262306330594644357, −6.12325317379840702370432527198, −5.52690715894799725912991265998, −4.38894900400379226973885475084, −4.13657478618381926099142727294, −3.35212496054430347862509421519, −1.93269917886883110924134406834, −1.02614237433189651646899691619, 0,
1.02614237433189651646899691619, 1.93269917886883110924134406834, 3.35212496054430347862509421519, 4.13657478618381926099142727294, 4.38894900400379226973885475084, 5.52690715894799725912991265998, 6.12325317379840702370432527198, 6.63173300142262306330594644357, 7.73253269809473547449052688065
