| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 8-s + 3·9-s − 6·11-s + 2·12-s + 16-s + 6·17-s + 3·18-s − 10·19-s − 6·22-s + 2·24-s − 9·25-s + 4·27-s + 32-s − 12·33-s + 6·34-s + 3·36-s − 10·38-s + 4·41-s − 12·43-s − 6·44-s + 2·48-s − 10·49-s − 9·50-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 − 1.80·11-s + 0.577·12-s + 1/4·16-s + 1.45·17-s + 0.707·18-s − 2.29·19-s − 1.27·22-s + 0.408·24-s − 9/5·25-s + 0.769·27-s + 0.176·32-s − 2.08·33-s + 1.02·34-s + 1/2·36-s − 1.62·38-s + 0.624·41-s − 1.82·43-s − 0.904·44-s + 0.288·48-s − 1.42·49-s − 1.27·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1107072 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1107072 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
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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
−7.949410618797364341592399601756, −7.33319367325295365011876797445, −7.29930015224798361544577008561, −6.28490963407970737599389441415, −6.21449897841157477524696518952, −5.57720553180084648062817612195, −5.03620516471027669405867234682, −4.68158569734292072473206781044, −4.07423093869793852668690105799, −3.54463186962642716865631176024, −3.23713315621654040660767256370, −2.34546937125578111204655036909, −2.31359724345954716585183902746, −1.50281149933669511104049990790, 0,
1.50281149933669511104049990790, 2.31359724345954716585183902746, 2.34546937125578111204655036909, 3.23713315621654040660767256370, 3.54463186962642716865631176024, 4.07423093869793852668690105799, 4.68158569734292072473206781044, 5.03620516471027669405867234682, 5.57720553180084648062817612195, 6.21449897841157477524696518952, 6.28490963407970737599389441415, 7.29930015224798361544577008561, 7.33319367325295365011876797445, 7.949410618797364341592399601756
