| L(s) = 1 | − 2·2-s + 2·4-s − 2·5-s − 2·7-s − 4·8-s + 4·10-s + 4·14-s + 8·16-s + 10·17-s + 4·19-s − 4·20-s + 8·23-s + 3·25-s − 4·28-s + 2·29-s − 4·31-s − 8·32-s − 20·34-s + 4·35-s − 8·37-s − 8·38-s + 8·40-s − 14·41-s − 4·43-s − 16·46-s + 10·47-s + 49-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.894·5-s − 0.755·7-s − 1.41·8-s + 1.26·10-s + 1.06·14-s + 2·16-s + 2.42·17-s + 0.917·19-s − 0.894·20-s + 1.66·23-s + 3/5·25-s − 0.755·28-s + 0.371·29-s − 0.718·31-s − 1.41·32-s − 3.42·34-s + 0.676·35-s − 1.31·37-s − 1.29·38-s + 1.26·40-s − 2.18·41-s − 0.609·43-s − 2.35·46-s + 1.45·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57836025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57836025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 10 T + 56 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 56 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 39 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 14 T + 128 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 87 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 92 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 18 T + 196 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T + 111 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 18 T + 203 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 22 T + 260 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 63 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 135 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 190 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 184 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 26 T + 351 T^{2} - 26 p T^{3} + p^{2} T^{4} \) |
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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
−7.54148415053992095658560073741, −7.48681871367425135466848299423, −7.30693820048572745049057502828, −6.79049618465863224590740014512, −6.51283018523610979781662079205, −5.93071805067082377411300814757, −5.70536095288057817996588194593, −5.42790696920699000623971358107, −4.77696576748326439747505972507, −4.76864451751183392411411144937, −3.73013629301501020589259344855, −3.61119893830739079532843018727, −3.23086089839306556243710949580, −2.87643957144702388216065273453, −2.79488583044377622843072655319, −1.62527163086152550526846431693, −1.31313546234294107033929749413, −0.966497683811036615818317136827, 0, 0,
0.966497683811036615818317136827, 1.31313546234294107033929749413, 1.62527163086152550526846431693, 2.79488583044377622843072655319, 2.87643957144702388216065273453, 3.23086089839306556243710949580, 3.61119893830739079532843018727, 3.73013629301501020589259344855, 4.76864451751183392411411144937, 4.77696576748326439747505972507, 5.42790696920699000623971358107, 5.70536095288057817996588194593, 5.93071805067082377411300814757, 6.51283018523610979781662079205, 6.79049618465863224590740014512, 7.30693820048572745049057502828, 7.48681871367425135466848299423, 7.54148415053992095658560073741
