| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 8-s + 3·9-s − 2·11-s + 2·12-s + 16-s − 12·17-s − 3·18-s − 8·19-s + 2·22-s − 2·24-s − 10·25-s + 4·27-s − 32-s − 4·33-s + 12·34-s + 3·36-s + 8·38-s + 12·41-s + 16·43-s − 2·44-s + 2·48-s − 10·49-s + 10·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.353·8-s + 9-s − 0.603·11-s + 0.577·12-s + 1/4·16-s − 2.91·17-s − 0.707·18-s − 1.83·19-s + 0.426·22-s − 0.408·24-s − 2·25-s + 0.769·27-s − 0.176·32-s − 0.696·33-s + 2.05·34-s + 1/2·36-s + 1.29·38-s + 1.87·41-s + 2.43·43-s − 0.301·44-s + 0.288·48-s − 1.42·49-s + 1.41·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139392 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139392 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−8.958855021658912251315559727512, −8.887891532747829374104895475794, −8.013104010237496671597514230187, −7.935843343791211954271432561876, −7.28527342169643413043846575990, −6.77506565559004391547494912024, −6.17174530290345741608508655802, −5.87020148568317205580403494022, −4.69333471332706230755932945396, −4.23690400616092916957233780303, −3.91425307940557450920960721713, −2.68572029653394044118459270403, −2.37545091357255938865951638025, −1.82843170642390821308972267128, 0,
1.82843170642390821308972267128, 2.37545091357255938865951638025, 2.68572029653394044118459270403, 3.91425307940557450920960721713, 4.23690400616092916957233780303, 4.69333471332706230755932945396, 5.87020148568317205580403494022, 6.17174530290345741608508655802, 6.77506565559004391547494912024, 7.28527342169643413043846575990, 7.935843343791211954271432561876, 8.013104010237496671597514230187, 8.887891532747829374104895475794, 8.958855021658912251315559727512
