| L(s) = 1 | − 2·5-s − 2·7-s + 4·13-s + 2·17-s − 12·19-s + 2·23-s + 3·25-s + 10·29-s − 2·31-s + 4·35-s + 2·37-s − 6·41-s − 4·43-s + 4·47-s − 3·49-s − 2·53-s − 2·59-s − 8·65-s − 14·67-s + 14·71-s − 4·73-s − 14·83-s − 4·85-s + 24·89-s − 8·91-s + 24·95-s + 8·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s + 1.10·13-s + 0.485·17-s − 2.75·19-s + 0.417·23-s + 3/5·25-s + 1.85·29-s − 0.359·31-s + 0.676·35-s + 0.328·37-s − 0.937·41-s − 0.609·43-s + 0.583·47-s − 3/7·49-s − 0.274·53-s − 0.260·59-s − 0.992·65-s − 1.71·67-s + 1.66·71-s − 0.468·73-s − 1.53·83-s − 0.433·85-s + 2.54·89-s − 0.838·91-s + 2.46·95-s + 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68558400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68558400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 81 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 73 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 80 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 89 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 120 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 14 T + 111 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 14 T + 141 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 132 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 197 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 24 T + 314 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 10 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.61387084565793261132960456666, −7.37029504934545388518216116759, −6.69411194860459284108667353567, −6.63014110093832571143974990674, −6.31479987379884985049418310361, −6.19191041308794249825864975807, −5.54560815260632512076194948020, −5.16705643027830345363843169042, −4.72541510923726691764931863939, −4.42158194518322993652260208234, −3.92694268542934718797704256799, −3.87185604955605536891333044258, −3.22552380492475594300314442181, −3.04899332776236357721850107714, −2.47616706420876862892647161764, −2.09612169728449294026630442397, −1.33065655145638882937826496227, −1.05685932111741774563708975689, 0, 0,
1.05685932111741774563708975689, 1.33065655145638882937826496227, 2.09612169728449294026630442397, 2.47616706420876862892647161764, 3.04899332776236357721850107714, 3.22552380492475594300314442181, 3.87185604955605536891333044258, 3.92694268542934718797704256799, 4.42158194518322993652260208234, 4.72541510923726691764931863939, 5.16705643027830345363843169042, 5.54560815260632512076194948020, 6.19191041308794249825864975807, 6.31479987379884985049418310361, 6.63014110093832571143974990674, 6.69411194860459284108667353567, 7.37029504934545388518216116759, 7.61387084565793261132960456666
