| L(s) = 1 | − 2-s − 3-s − 4-s + 6-s − 4·7-s + 3·8-s + 9-s + 4·11-s + 12-s + 2·13-s + 4·14-s − 16-s − 18-s + 4·19-s + 4·21-s − 4·22-s − 4·23-s − 3·24-s − 5·25-s − 2·26-s − 27-s + 4·28-s + 4·31-s − 5·32-s − 4·33-s − 36-s − 8·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.51·7-s + 1.06·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s + 0.554·13-s + 1.06·14-s − 1/4·16-s − 0.235·18-s + 0.917·19-s + 0.872·21-s − 0.852·22-s − 0.834·23-s − 0.612·24-s − 25-s − 0.392·26-s − 0.192·27-s + 0.755·28-s + 0.718·31-s − 0.883·32-s − 0.696·33-s − 1/6·36-s − 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−9.846817875642606686071714436912, −9.085950493171175712008401589868, −8.222977725542171544239947078177, −7.07777719442002714062267999051, −6.39243185629308779636289782974, −5.48817358399049682954766371208, −4.17171970899973202542642184176, −3.41647740680311471448286449917, −1.42437354645913391865975562541, 0,
1.42437354645913391865975562541, 3.41647740680311471448286449917, 4.17171970899973202542642184176, 5.48817358399049682954766371208, 6.39243185629308779636289782974, 7.07777719442002714062267999051, 8.222977725542171544239947078177, 9.085950493171175712008401589868, 9.846817875642606686071714436912
