L(s) = 1 | + 2·3-s + 9-s − 4·11-s − 2·25-s − 4·27-s − 8·33-s − 6·49-s + 12·59-s − 20·73-s − 4·75-s − 11·81-s − 12·83-s − 12·97-s − 4·99-s + 12·107-s − 10·121-s + 127-s + 131-s + 137-s + 139-s − 12·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1/3·9-s − 1.20·11-s − 2/5·25-s − 0.769·27-s − 1.39·33-s − 6/7·49-s + 1.56·59-s − 2.34·73-s − 0.461·75-s − 1.22·81-s − 1.31·83-s − 1.21·97-s − 0.402·99-s + 1.16·107-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.989·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 150 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 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
−8.255625300806254601557168078213, −7.77518155369436790721434568868, −7.50910262165143853067485435520, −6.96670093342373572471155333070, −6.46697044761023463065495811788, −5.72176553148235727766177367236, −5.52944593450315156828442679052, −4.87485435778759102409394582756, −4.29156900530558683160437840953, −3.77494466900218907857664466801, −3.15146764745142520282208548511, −2.70087550689168111438602316121, −2.21881059253034472843226343471, −1.42267829556748512158345581079, 0,
1.42267829556748512158345581079, 2.21881059253034472843226343471, 2.70087550689168111438602316121, 3.15146764745142520282208548511, 3.77494466900218907857664466801, 4.29156900530558683160437840953, 4.87485435778759102409394582756, 5.52944593450315156828442679052, 5.72176553148235727766177367236, 6.46697044761023463065495811788, 6.96670093342373572471155333070, 7.50910262165143853067485435520, 7.77518155369436790721434568868, 8.255625300806254601557168078213