| L(s) = 1 | + 2·2-s + 2·4-s − 2·5-s + 9-s − 4·10-s − 4·13-s − 4·16-s + 4·17-s + 2·18-s − 4·20-s − 7·25-s − 8·26-s + 6·29-s − 8·32-s + 8·34-s + 2·36-s − 14·37-s + 20·41-s − 2·45-s − 5·49-s − 14·50-s − 8·52-s + 8·53-s + 12·58-s − 20·61-s − 8·64-s + 8·65-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.894·5-s + 1/3·9-s − 1.26·10-s − 1.10·13-s − 16-s + 0.970·17-s + 0.471·18-s − 0.894·20-s − 7/5·25-s − 1.56·26-s + 1.11·29-s − 1.41·32-s + 1.37·34-s + 1/3·36-s − 2.30·37-s + 3.12·41-s − 0.298·45-s − 5/7·49-s − 1.97·50-s − 1.10·52-s + 1.09·53-s + 1.57·58-s − 2.56·61-s − 64-s + 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 318096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 318096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
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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
−8.497533279987248232679778769473, −7.79027625446740945229498901534, −7.52539857035144694302700552711, −7.28827588046584127313119350976, −6.46487870715497229983731599323, −6.15895661640491039854976545718, −5.48295326491665310465200067843, −5.14885571821443324588162204875, −4.54291964872185891125664944401, −4.04609810907902673146942974166, −3.77257881491072191084851286374, −2.93657054326743540839942565323, −2.58057258752762911906859617259, −1.52773270357203329962095932751, 0,
1.52773270357203329962095932751, 2.58057258752762911906859617259, 2.93657054326743540839942565323, 3.77257881491072191084851286374, 4.04609810907902673146942974166, 4.54291964872185891125664944401, 5.14885571821443324588162204875, 5.48295326491665310465200067843, 6.15895661640491039854976545718, 6.46487870715497229983731599323, 7.28827588046584127313119350976, 7.52539857035144694302700552711, 7.79027625446740945229498901534, 8.497533279987248232679778769473
