| L(s) = 1 | + 2-s − 54·3-s − 37·4-s − 194·5-s − 54·6-s − 418·7-s + 53·8-s + 2.18e3·9-s − 194·10-s + 2.66e3·11-s + 1.99e3·12-s − 1.32e4·13-s − 418·14-s + 1.04e4·15-s − 1.47e4·16-s − 1.02e4·17-s + 2.18e3·18-s + 1.41e4·19-s + 7.17e3·20-s + 2.25e4·21-s + 2.66e3·22-s − 1.36e4·23-s − 2.86e3·24-s − 8.52e4·25-s − 1.32e4·26-s − 7.87e4·27-s + 1.54e4·28-s + ⋯ |
| L(s) = 1 | + 0.0883·2-s − 1.15·3-s − 0.289·4-s − 0.694·5-s − 0.102·6-s − 0.460·7-s + 0.0365·8-s + 9-s − 0.0613·10-s + 0.603·11-s + 0.333·12-s − 1.67·13-s − 0.0407·14-s + 0.801·15-s − 0.903·16-s − 0.506·17-s + 0.0883·18-s + 0.474·19-s + 0.200·20-s + 0.531·21-s + 0.0533·22-s − 0.234·23-s − 0.0422·24-s − 1.09·25-s − 0.147·26-s − 0.769·27-s + 0.133·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + 19 p T^{2} - p^{7} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 194 T + 122882 T^{2} + 194 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 418 T + 1305774 T^{2} + 418 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 13246 T + 169046010 T^{2} + 13246 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 10256 T + 839902430 T^{2} + 10256 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 14196 T + 610166774 T^{2} - 14196 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 13666 T + 5939145158 T^{2} + 13666 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 528 p T + 30012198502 T^{2} - 528 p^{8} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 48040 T + 55594799550 T^{2} + 48040 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 274092 T + 171168601790 T^{2} - 274092 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 755836 T + 339658819958 T^{2} - 755836 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 1704096 T + 1258258595846 T^{2} + 1704096 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1182094 T + 965730056342 T^{2} + 1182094 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2156394 T + 3355666109890 T^{2} + 2156394 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 927332 T + 3919039493894 T^{2} - 927332 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1061994 T + 60938203298 T^{2} + 1061994 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 48656 p T + 14776996218054 T^{2} + 48656 p^{8} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5495514 T + 25737621036694 T^{2} + 5495514 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5450812 T + 26436565990902 T^{2} + 5450812 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1536590 T + 38585225535486 T^{2} + 1536590 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8850888 T + 59626634719558 T^{2} - 8850888 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6810132 T + 98894344907446 T^{2} - 6810132 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 9897376 T + 135461286702270 T^{2} - 9897376 p^{7} T^{3} + p^{14} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.76083765316836704468180056454, −14.54438416465071341997684971003, −13.25479144991627693396392748898, −13.18543594500227010929755863959, −12.17595014991538835575592129113, −11.68867612192354316618897898708, −11.46489005783242147271280757105, −10.50273010917865603031675098429, −9.586179327422637358825978524208, −9.480695254696126452903927650318, −8.132485007663403948096680305629, −7.40711715227314437468066439237, −6.69127306155393940452628761948, −6.00641169289854054779262943886, −4.78335484458172153954764865401, −4.51741988891698756997874784254, −3.26307801700169155058240601211, −1.75715288048953095286512485367, 0, 0,
1.75715288048953095286512485367, 3.26307801700169155058240601211, 4.51741988891698756997874784254, 4.78335484458172153954764865401, 6.00641169289854054779262943886, 6.69127306155393940452628761948, 7.40711715227314437468066439237, 8.132485007663403948096680305629, 9.480695254696126452903927650318, 9.586179327422637358825978524208, 10.50273010917865603031675098429, 11.46489005783242147271280757105, 11.68867612192354316618897898708, 12.17595014991538835575592129113, 13.18543594500227010929755863959, 13.25479144991627693396392748898, 14.54438416465071341997684971003, 14.76083765316836704468180056454
