| L(s) = 1 | + 2·3-s + 2·5-s − 2·7-s + 2·9-s + 4·11-s + 6·13-s + 4·15-s − 2·19-s − 4·21-s − 8·23-s − 2·25-s + 6·27-s − 4·31-s + 8·33-s − 4·35-s + 12·39-s − 8·41-s − 4·43-s + 4·45-s − 12·47-s + 3·49-s − 20·53-s + 8·55-s − 4·57-s + 14·59-s + 18·61-s − 4·63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s − 0.755·7-s + 2/3·9-s + 1.20·11-s + 1.66·13-s + 1.03·15-s − 0.458·19-s − 0.872·21-s − 1.66·23-s − 2/5·25-s + 1.15·27-s − 0.718·31-s + 1.39·33-s − 0.676·35-s + 1.92·39-s − 1.24·41-s − 0.609·43-s + 0.596·45-s − 1.75·47-s + 3/7·49-s − 2.74·53-s + 1.07·55-s − 0.529·57-s + 1.82·59-s + 2.30·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50176 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.316602582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.316602582\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 14 T + 162 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 18 T + 198 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 210 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 238 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−12.69184392290246865848785566473, −12.05972195931707119663777641652, −11.47458123413300220597019137645, −11.14940950384920267977657764595, −10.13066451433255203489171863316, −10.12554122039553452308402962481, −9.572373527065341565451768546361, −9.042883762049257040898170081217, −8.524978190737606320675831669004, −8.356816000659126457280838882807, −7.62017593248943357078401779319, −6.71752546290169180092862320997, −6.36402485764266418077390521555, −6.11868239118300673563659650682, −5.23651453418504227026658067374, −4.29668658677344589822753768806, −3.48053718942393294634255816722, −3.45664727824938054196263640486, −2.15948301807031587246528686001, −1.57060045378259883863154228778,
1.57060045378259883863154228778, 2.15948301807031587246528686001, 3.45664727824938054196263640486, 3.48053718942393294634255816722, 4.29668658677344589822753768806, 5.23651453418504227026658067374, 6.11868239118300673563659650682, 6.36402485764266418077390521555, 6.71752546290169180092862320997, 7.62017593248943357078401779319, 8.356816000659126457280838882807, 8.524978190737606320675831669004, 9.042883762049257040898170081217, 9.572373527065341565451768546361, 10.12554122039553452308402962481, 10.13066451433255203489171863316, 11.14940950384920267977657764595, 11.47458123413300220597019137645, 12.05972195931707119663777641652, 12.69184392290246865848785566473
