| L(s) = 1 | − 4·2-s + 8·4-s − 8·8-s − 5·9-s + 8·13-s − 4·16-s − 2·17-s + 20·18-s − 9·25-s − 32·26-s + 32·32-s + 8·34-s − 40·36-s − 12·43-s + 16·47-s − 10·49-s + 36·50-s + 64·52-s − 12·53-s + 10·59-s − 64·64-s − 14·67-s − 16·68-s + 40·72-s + 16·81-s − 12·83-s + 48·86-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4·4-s − 2.82·8-s − 5/3·9-s + 2.21·13-s − 16-s − 0.485·17-s + 4.71·18-s − 9/5·25-s − 6.27·26-s + 5.65·32-s + 1.37·34-s − 6.66·36-s − 1.82·43-s + 2.33·47-s − 1.42·49-s + 5.09·50-s + 8.87·52-s − 1.64·53-s + 1.30·59-s − 8·64-s − 1.71·67-s − 1.94·68-s + 4.71·72-s + 16/9·81-s − 1.31·83-s + 5.17·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34969 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−10.03550909718107888433464868208, −9.369611094308164202184756440285, −9.018955124365094927616403290559, −8.603539619290756001226038948684, −8.263155997010663128403798948932, −7.907366117668739819170581630667, −7.29774796427450513185159809590, −6.36261389471308870138602900888, −6.22323910317192896404427293386, −5.32604921703423297753132504909, −4.24329958875835728994074784592, −3.36204123451452550822417446571, −2.23424739606738503289626266310, −1.35872004735313563095158147020, 0,
1.35872004735313563095158147020, 2.23424739606738503289626266310, 3.36204123451452550822417446571, 4.24329958875835728994074784592, 5.32604921703423297753132504909, 6.22323910317192896404427293386, 6.36261389471308870138602900888, 7.29774796427450513185159809590, 7.907366117668739819170581630667, 8.263155997010663128403798948932, 8.603539619290756001226038948684, 9.018955124365094927616403290559, 9.369611094308164202184756440285, 10.03550909718107888433464868208
