| L(s) = 1 | − 4·2-s − 2·3-s − 4-s + 16·5-s + 8·6-s + 22·7-s + 40·8-s − 24·9-s − 64·10-s + 30·11-s + 2·12-s − 22·13-s − 88·14-s − 32·15-s − 79·16-s − 104·17-s + 96·18-s + 10·19-s − 16·20-s − 44·21-s − 120·22-s − 80·24-s + 89·25-s + 88·26-s + 50·27-s − 22·28-s − 146·29-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.384·3-s − 1/8·4-s + 1.43·5-s + 0.544·6-s + 1.18·7-s + 1.76·8-s − 8/9·9-s − 2.02·10-s + 0.822·11-s + 0.0481·12-s − 0.469·13-s − 1.67·14-s − 0.550·15-s − 1.23·16-s − 1.48·17-s + 1.25·18-s + 0.120·19-s − 0.178·20-s − 0.457·21-s − 1.16·22-s − 0.680·24-s + 0.711·25-s + 0.663·26-s + 0.356·27-s − 0.148·28-s − 0.934·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 279841 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 279841 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 23 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + p^{2} T + 17 T^{2} + p^{5} T^{3} + p^{6} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 2 T + 28 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 16 T + 167 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 22 T + 804 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 30 T + 2740 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 22 T + 627 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 104 T + 10502 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 10 T + 6540 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 146 T + 39407 T^{2} + 146 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 172 T + 63090 T^{2} + 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T - 9282 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 270 T + 151735 T^{2} + 270 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 500 T + 207642 T^{2} + 500 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 258 T + 4220 p T^{2} + 258 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 196 T - 33349 T^{2} + 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 614 T + 247460 T^{2} + 614 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 100 T + 419499 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 36 T + 356462 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 290 T + 88172 T^{2} + 290 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 318 T + 435815 T^{2} - 318 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 212 T + 421782 T^{2} - 212 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1476 T + 1577626 T^{2} - 1476 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1308 T + 893491 T^{2} + 1308 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 200 T + 1595079 T^{2} - 200 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.16046883289293437403870827166, −9.568771795061561506313682611142, −9.225746851066610406321131377292, −9.136609959410572516232461787681, −8.538942135627431483533937736312, −8.160640705044009513444323313412, −7.81372111262827861094583859727, −7.12248885339779831800563285350, −6.48337562813275136329375205817, −6.26136903590134623644038494051, −5.38869106762302284024306731161, −5.11735994326322244027496324253, −4.73815827400505116680843656878, −4.06266092931603788389343627597, −3.24028782975744983405208301317, −2.22650075762756716185071121546, −1.64545778910554448954474261357, −1.41212313178524008269997392009, 0, 0,
1.41212313178524008269997392009, 1.64545778910554448954474261357, 2.22650075762756716185071121546, 3.24028782975744983405208301317, 4.06266092931603788389343627597, 4.73815827400505116680843656878, 5.11735994326322244027496324253, 5.38869106762302284024306731161, 6.26136903590134623644038494051, 6.48337562813275136329375205817, 7.12248885339779831800563285350, 7.81372111262827861094583859727, 8.160640705044009513444323313412, 8.538942135627431483533937736312, 9.136609959410572516232461787681, 9.225746851066610406321131377292, 9.568771795061561506313682611142, 10.16046883289293437403870827166
