| L(s) = 1 | − 2·2-s + 2·4-s + 2·5-s − 4·7-s − 5·9-s − 4·10-s + 11-s + 8·14-s − 4·16-s + 10·18-s + 4·20-s − 2·22-s − 7·25-s − 8·28-s + 8·32-s − 8·35-s − 10·36-s + 6·37-s − 12·43-s + 2·44-s − 10·45-s − 2·49-s + 14·50-s − 12·53-s + 2·55-s + 20·63-s − 8·64-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.894·5-s − 1.51·7-s − 5/3·9-s − 1.26·10-s + 0.301·11-s + 2.13·14-s − 16-s + 2.35·18-s + 0.894·20-s − 0.426·22-s − 7/5·25-s − 1.51·28-s + 1.41·32-s − 1.35·35-s − 5/3·36-s + 0.986·37-s − 1.82·43-s + 0.301·44-s − 1.49·45-s − 2/7·49-s + 1.97·50-s − 1.64·53-s + 0.269·55-s + 2.51·63-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 11 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−10.03550909718107888433464868208, −10.01198035595578727343029497253, −9.555446494172697123197364265116, −8.979273038504653746902140082359, −8.603539619290756001226038948684, −7.961176778048713923667541484628, −7.39347658072832330462797120078, −6.36261389471308870138602900888, −6.34194634679926922702796543533, −5.68468142602311525135152832168, −4.75527439375896786693668259973, −3.52526734376529108704825072242, −2.84374992801449552338801496278, −1.87273627454943730491171614963, 0,
1.87273627454943730491171614963, 2.84374992801449552338801496278, 3.52526734376529108704825072242, 4.75527439375896786693668259973, 5.68468142602311525135152832168, 6.34194634679926922702796543533, 6.36261389471308870138602900888, 7.39347658072832330462797120078, 7.961176778048713923667541484628, 8.603539619290756001226038948684, 8.979273038504653746902140082359, 9.555446494172697123197364265116, 10.01198035595578727343029497253, 10.03550909718107888433464868208
