L(s) = 1 | − 3-s − 2·5-s + 7-s + 9-s − 4·11-s − 2·13-s + 2·15-s − 6·17-s − 4·19-s − 21-s − 25-s − 27-s − 2·29-s + 4·33-s − 2·35-s + 6·37-s + 2·39-s + 2·41-s + 4·43-s − 2·45-s + 49-s + 6·51-s + 6·53-s + 8·55-s + 4·57-s − 12·59-s − 2·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.516·15-s − 1.45·17-s − 0.917·19-s − 0.218·21-s − 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.696·33-s − 0.338·35-s + 0.986·37-s + 0.320·39-s + 0.312·41-s + 0.609·43-s − 0.298·45-s + 1/7·49-s + 0.840·51-s + 0.824·53-s + 1.07·55-s + 0.529·57-s − 1.56·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−11.04847280565543144871395360807, −10.51953132809604428367636932730, −9.198895146278336601365818133221, −8.068098981215555715874277637796, −7.38325958034671995701748313853, −6.19643457342138342025418009798, −4.94701466488767178340177905762, −4.13981751462080886088964913424, −2.39234840548789475685588947322, 0,
2.39234840548789475685588947322, 4.13981751462080886088964913424, 4.94701466488767178340177905762, 6.19643457342138342025418009798, 7.38325958034671995701748313853, 8.068098981215555715874277637796, 9.198895146278336601365818133221, 10.51953132809604428367636932730, 11.04847280565543144871395360807