| L(s) = 1 | + 2-s − 11·4-s + 10·5-s − 4·7-s − 15·8-s + 10·10-s + 22·11-s − 90·13-s − 4·14-s + 61·16-s + 16·17-s − 170·19-s − 110·20-s + 22·22-s + 124·23-s + 75·25-s − 90·26-s + 44·28-s + 158·29-s + 60·31-s + 89·32-s + 16·34-s − 40·35-s − 372·37-s − 170·38-s − 150·40-s − 38·41-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 1.37·4-s + 0.894·5-s − 0.215·7-s − 0.662·8-s + 0.316·10-s + 0.603·11-s − 1.92·13-s − 0.0763·14-s + 0.953·16-s + 0.228·17-s − 2.05·19-s − 1.22·20-s + 0.213·22-s + 1.12·23-s + 3/5·25-s − 0.678·26-s + 0.296·28-s + 1.01·29-s + 0.347·31-s + 0.491·32-s + 0.0807·34-s − 0.193·35-s − 1.65·37-s − 0.725·38-s − 0.592·40-s − 0.144·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 p^{2} T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 622 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 90 T + 6402 T^{2} + 90 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 16 T + 1662 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 170 T + 994 p T^{2} + 170 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 124 T + 12878 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 158 T + 50106 T^{2} - 158 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 60 T + 59870 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 372 T + 82590 T^{2} + 372 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 38 T + 37410 T^{2} + 38 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 p T + 160230 T^{2} + 12 p^{4} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 224 T + 466 p T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 472 T + 190182 T^{2} + 472 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 248 T + 181334 T^{2} + 248 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 72 T - 107850 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 744 T + 738822 T^{2} + 744 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2060 T + 1768494 T^{2} + 2060 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 486 T + 822786 T^{2} + 486 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 642 T + 691166 T^{2} - 642 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 286 T + 392750 T^{2} - 286 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 244 T + 1355190 T^{2} + 244 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 168 T + 1053870 T^{2} + 168 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.13245662428900951426411017863, −9.948163573622847215252595845269, −9.371755726495176599387817081107, −9.077248386688530555116035978633, −8.527017468236425571268673193362, −8.383063868392147302400508649267, −7.54440657040370455372181697887, −7.00130698532024725838238632290, −6.41797549725975424547959293615, −6.27472499112418587661656578341, −5.33945680527724516925475748585, −5.00070831600848173878349309545, −4.57726716946648769513464080531, −4.31975899509896448404227576913, −3.21493357478241499562303704031, −3.01925994101984347803007631576, −1.99804556965075284609601186932, −1.41939565351603278287198016777, 0, 0,
1.41939565351603278287198016777, 1.99804556965075284609601186932, 3.01925994101984347803007631576, 3.21493357478241499562303704031, 4.31975899509896448404227576913, 4.57726716946648769513464080531, 5.00070831600848173878349309545, 5.33945680527724516925475748585, 6.27472499112418587661656578341, 6.41797549725975424547959293615, 7.00130698532024725838238632290, 7.54440657040370455372181697887, 8.383063868392147302400508649267, 8.527017468236425571268673193362, 9.077248386688530555116035978633, 9.371755726495176599387817081107, 9.948163573622847215252595845269, 10.13245662428900951426411017863
