L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s − 2·9-s − 8·13-s − 2·14-s + 16-s + 2·18-s + 4·19-s − 10·25-s + 8·26-s + 2·28-s − 12·29-s − 32-s − 2·36-s − 4·38-s + 12·41-s + 16·43-s + 3·49-s + 10·50-s − 8·52-s − 2·56-s + 12·58-s − 4·63-s + 64-s − 8·67-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s − 2/3·9-s − 2.21·13-s − 0.534·14-s + 1/4·16-s + 0.471·18-s + 0.917·19-s − 2·25-s + 1.56·26-s + 0.377·28-s − 2.22·29-s − 0.176·32-s − 1/3·36-s − 0.648·38-s + 1.87·41-s + 2.43·43-s + 3/7·49-s + 1.41·50-s − 1.10·52-s − 0.267·56-s + 1.57·58-s − 0.503·63-s + 1/8·64-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 829472 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 829472 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8185683604\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8185683604\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_2$ | \( 1 + p T^{2} \) |
good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.964944662527870856026164037982, −7.64178149897623079839720524725, −7.57571100088867902110310233811, −7.28614091898437125853646778423, −6.48583845948574201695144290108, −5.81994722151176032382577376975, −5.57928681742950427486583645839, −5.25222358335302626915983382813, −4.45632876697515521773485306871, −4.09738071226881210588087278636, −3.39198469441863826812749201545, −2.50137257751084499884008386915, −2.37043392826247813268923532045, −1.61447913787792106227712636889, −0.46915074201749145746453250386,
0.46915074201749145746453250386, 1.61447913787792106227712636889, 2.37043392826247813268923532045, 2.50137257751084499884008386915, 3.39198469441863826812749201545, 4.09738071226881210588087278636, 4.45632876697515521773485306871, 5.25222358335302626915983382813, 5.57928681742950427486583645839, 5.81994722151176032382577376975, 6.48583845948574201695144290108, 7.28614091898437125853646778423, 7.57571100088867902110310233811, 7.64178149897623079839720524725, 7.964944662527870856026164037982