| L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s + 9-s + 2·10-s + 2·13-s + 16-s − 12·17-s + 18-s + 2·20-s + 3·25-s + 2·26-s + 12·29-s + 32-s − 12·34-s + 36-s − 20·37-s + 2·40-s − 12·41-s + 2·45-s − 14·49-s + 3·50-s + 2·52-s − 20·53-s + 12·58-s − 4·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.554·13-s + 1/4·16-s − 2.91·17-s + 0.235·18-s + 0.447·20-s + 3/5·25-s + 0.392·26-s + 2.22·29-s + 0.176·32-s − 2.05·34-s + 1/6·36-s − 3.28·37-s + 0.316·40-s − 1.87·41-s + 0.298·45-s − 2·49-s + 0.424·50-s + 0.277·52-s − 2.74·53-s + 1.57·58-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−7.83417942869426066771961040753, −7.03582983327921838179279968038, −6.57950396679982686721830292017, −6.57267675826487089075525795107, −6.33947776642796638344079621345, −5.43160779320744251383504417132, −5.06664273077784135902280870302, −4.53496762516360346181339328954, −4.50539658406620147914202325372, −3.42325323576864435489079625967, −3.26654328534844814185904778325, −2.46147122738666541505186218774, −1.80910327227356048364459418629, −1.55977141690761434296396620452, 0,
1.55977141690761434296396620452, 1.80910327227356048364459418629, 2.46147122738666541505186218774, 3.26654328534844814185904778325, 3.42325323576864435489079625967, 4.50539658406620147914202325372, 4.53496762516360346181339328954, 5.06664273077784135902280870302, 5.43160779320744251383504417132, 6.33947776642796638344079621345, 6.57267675826487089075525795107, 6.57950396679982686721830292017, 7.03582983327921838179279968038, 7.83417942869426066771961040753
