| L(s) = 1 | − 2·2-s − 10·3-s − 11·4-s − 10·5-s + 20·6-s − 16·7-s + 36·8-s + 39·9-s + 20·10-s − 26·11-s + 110·12-s − 26·13-s + 32·14-s + 100·15-s + 61·16-s + 60·17-s − 78·18-s − 220·19-s + 110·20-s + 160·21-s + 52·22-s + 52·23-s − 360·24-s − 143·25-s + 52·26-s − 50·27-s + 176·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.92·3-s − 1.37·4-s − 0.894·5-s + 1.36·6-s − 0.863·7-s + 1.59·8-s + 13/9·9-s + 0.632·10-s − 0.712·11-s + 2.64·12-s − 0.554·13-s + 0.610·14-s + 1.72·15-s + 0.953·16-s + 0.856·17-s − 1.02·18-s − 2.65·19-s + 1.22·20-s + 1.66·21-s + 0.503·22-s + 0.471·23-s − 3.06·24-s − 1.14·25-s + 0.392·26-s − 0.356·27-s + 1.18·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 29 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 15 T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 10 T + 61 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 p T + 243 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 16 T + 550 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 26 T + 93 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 p T + 19 p^{2} T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 60 T + 10078 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 220 T + 23770 T^{2} + 220 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 52 T + 20402 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 294 T + 2363 p T^{2} + 294 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 312 T + 119370 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 40 T + 100154 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 322 T + 126453 T^{2} + 322 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 130 T + 126173 T^{2} + 130 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 1002 T + 518987 T^{2} - 1002 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 900 T + 490250 T^{2} + 900 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 948 T + 615270 T^{2} + 948 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 320 T + 158614 T^{2} - 320 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 660 T + 822410 T^{2} + 660 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 648 T + 63810 T^{2} - 648 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 258 T + 770157 T^{2} - 258 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1212 T + 1502618 T^{2} - 1212 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 760 T + 1009370 T^{2} - 760 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 24 T + 1157322 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−16.46976932620168704908211552687, −16.33833750590260599028789669791, −14.99362885371431321175005214260, −14.90157465963589865550929850725, −13.49720619638542510200395714815, −13.12577067531635209909531596123, −12.36325042905113069824494992125, −12.06699832388332366568260854880, −10.89075172801174481768107751876, −10.72109520463557624017405549539, −9.862476516189287789103102992686, −9.233686056546093846184361494433, −8.235515118369237279416647585400, −7.66321302669574866883891850884, −6.45363169599506477439245978795, −5.62777663248966123170236433844, −4.79240736397582967434846963710, −3.92457225982003136766281733809, 0, 0,
3.92457225982003136766281733809, 4.79240736397582967434846963710, 5.62777663248966123170236433844, 6.45363169599506477439245978795, 7.66321302669574866883891850884, 8.235515118369237279416647585400, 9.233686056546093846184361494433, 9.862476516189287789103102992686, 10.72109520463557624017405549539, 10.89075172801174481768107751876, 12.06699832388332366568260854880, 12.36325042905113069824494992125, 13.12577067531635209909531596123, 13.49720619638542510200395714815, 14.90157465963589865550929850725, 14.99362885371431321175005214260, 16.33833750590260599028789669791, 16.46976932620168704908211552687
