L(s) = 1 | − 2-s − 4-s − 6·5-s + 3·8-s − 2·9-s + 6·10-s − 2·13-s − 16-s − 2·17-s + 2·18-s + 6·20-s + 17·25-s + 2·26-s − 4·29-s − 5·32-s + 2·34-s + 2·36-s − 3·37-s − 18·40-s + 6·41-s + 12·45-s − 12·49-s − 17·50-s + 2·52-s − 2·53-s + 4·58-s + 13·61-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 2.68·5-s + 1.06·8-s − 2/3·9-s + 1.89·10-s − 0.554·13-s − 1/4·16-s − 0.485·17-s + 0.471·18-s + 1.34·20-s + 17/5·25-s + 0.392·26-s − 0.742·29-s − 0.883·32-s + 0.342·34-s + 1/3·36-s − 0.493·37-s − 2.84·40-s + 0.937·41-s + 1.78·45-s − 1.71·49-s − 2.40·50-s + 0.277·52-s − 0.274·53-s + 0.525·58-s + 1.66·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 801296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 801296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 12 T + p T^{2} ) \) |
| 821 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 18 T + p T^{2} ) \) |
good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−8.109452997284717824152805619958, −7.78298345038900784898277489171, −7.24633553027470904809371779290, −7.08433560847438483914264670758, −6.42347635474255981727883493972, −5.66684588687903998607838958235, −5.14682960471281176171593221690, −4.60152788709140368813457229098, −4.25503985656433918406748871681, −3.80940509531773130113755015009, −3.36767316670945332664993146425, −2.73834667638012302830935944542, −1.75718417202665453353272778132, −0.59384375226056095779524050358, 0,
0.59384375226056095779524050358, 1.75718417202665453353272778132, 2.73834667638012302830935944542, 3.36767316670945332664993146425, 3.80940509531773130113755015009, 4.25503985656433918406748871681, 4.60152788709140368813457229098, 5.14682960471281176171593221690, 5.66684588687903998607838958235, 6.42347635474255981727883493972, 7.08433560847438483914264670758, 7.24633553027470904809371779290, 7.78298345038900784898277489171, 8.109452997284717824152805619958