| L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 2·6-s − 6·7-s + 9-s − 4·11-s − 2·12-s + 12·14-s − 4·16-s − 4·17-s − 2·18-s + 6·21-s + 8·22-s − 27-s − 12·28-s + 8·32-s + 4·33-s + 8·34-s + 2·36-s − 12·42-s + 2·43-s − 8·44-s + 4·48-s + 13·49-s + 4·51-s + 8·53-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s − 2.26·7-s + 1/3·9-s − 1.20·11-s − 0.577·12-s + 3.20·14-s − 16-s − 0.970·17-s − 0.471·18-s + 1.30·21-s + 1.70·22-s − 0.192·27-s − 2.26·28-s + 1.41·32-s + 0.696·33-s + 1.37·34-s + 1/3·36-s − 1.85·42-s + 0.304·43-s − 1.20·44-s + 0.577·48-s + 13/7·49-s + 0.560·51-s + 1.09·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.938558667143995354006242020404, −8.105315307318281005924759006469, −7.965886472126962800720761127653, −6.96660723932071234204269144809, −6.96141726621690601574492077524, −6.59754059703926529506773066348, −5.87930303915993667052597404250, −5.45352832460903594958062456545, −4.78985931636144963960080177818, −4.01465426298836833468759021488, −3.48045838663060024632885892599, −2.58914107657822994764341775063, −2.23186561512245260821665501913, −0.77482341387654784778243330666, 0,
0.77482341387654784778243330666, 2.23186561512245260821665501913, 2.58914107657822994764341775063, 3.48045838663060024632885892599, 4.01465426298836833468759021488, 4.78985931636144963960080177818, 5.45352832460903594958062456545, 5.87930303915993667052597404250, 6.59754059703926529506773066348, 6.96141726621690601574492077524, 6.96660723932071234204269144809, 7.965886472126962800720761127653, 8.105315307318281005924759006469, 8.938558667143995354006242020404
