| L(s) = 1 | − 2·3-s − 2·5-s + 2·7-s − 4·11-s + 4·15-s − 8·19-s − 4·21-s + 2·23-s + 3·25-s + 2·27-s − 4·31-s + 8·33-s − 4·35-s − 4·37-s + 4·41-s − 14·43-s + 10·47-s − 8·49-s + 16·53-s + 8·55-s + 16·57-s − 8·59-s + 4·61-s − 18·67-s − 4·69-s − 4·71-s − 8·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 0.755·7-s − 1.20·11-s + 1.03·15-s − 1.83·19-s − 0.872·21-s + 0.417·23-s + 3/5·25-s + 0.384·27-s − 0.718·31-s + 1.39·33-s − 0.676·35-s − 0.657·37-s + 0.624·41-s − 2.13·43-s + 1.45·47-s − 8/7·49-s + 2.19·53-s + 1.07·55-s + 2.11·57-s − 1.04·59-s + 0.512·61-s − 2.19·67-s − 0.481·69-s − 0.474·71-s − 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 14 T + 132 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 116 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 16 T + 158 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 122 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 18 T + 212 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 172 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.391738150394117520099174674125, −8.994185137357339770910940421061, −8.415944518628747317833252377998, −8.350748424080388590855313750503, −7.918280088936170377980472063450, −7.36927325093451665051001213038, −6.91089702743292962023004224971, −6.75435728538014509521987366642, −5.88369310702907686535844344730, −5.69423850032308280235077213487, −5.33118459531553052357886616137, −4.81110401028574691127705389576, −4.21718546521976294794410235524, −4.19837453503013577458556286505, −3.18634972588879378366548622359, −2.79632342734316949980518488179, −2.04343773812811076177872639546, −1.34206342860404764783195790078, 0, 0,
1.34206342860404764783195790078, 2.04343773812811076177872639546, 2.79632342734316949980518488179, 3.18634972588879378366548622359, 4.19837453503013577458556286505, 4.21718546521976294794410235524, 4.81110401028574691127705389576, 5.33118459531553052357886616137, 5.69423850032308280235077213487, 5.88369310702907686535844344730, 6.75435728538014509521987366642, 6.91089702743292962023004224971, 7.36927325093451665051001213038, 7.918280088936170377980472063450, 8.350748424080388590855313750503, 8.415944518628747317833252377998, 8.994185137357339770910940421061, 9.391738150394117520099174674125
