| L(s) = 1 | − 8·2-s + 48·4-s + 54·5-s − 74·7-s − 256·8-s − 432·10-s + 78·11-s − 1.10e3·13-s + 592·14-s + 1.28e3·16-s + 492·17-s − 1.64e3·19-s + 2.59e3·20-s − 624·22-s + 5.53e3·23-s − 3.19e3·25-s + 8.84e3·26-s − 3.55e3·28-s + 3.89e3·29-s − 4.71e3·31-s − 6.14e3·32-s − 3.93e3·34-s − 3.99e3·35-s − 4.79e3·37-s + 1.31e4·38-s − 1.38e4·40-s − 1.53e4·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.965·5-s − 0.570·7-s − 1.41·8-s − 1.36·10-s + 0.194·11-s − 1.81·13-s + 0.807·14-s + 5/4·16-s + 0.412·17-s − 1.04·19-s + 1.44·20-s − 0.274·22-s + 2.18·23-s − 1.02·25-s + 2.56·26-s − 0.856·28-s + 0.859·29-s − 0.881·31-s − 1.06·32-s − 0.583·34-s − 0.551·35-s − 0.575·37-s + 1.47·38-s − 1.36·40-s − 1.42·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 54 T + 1223 p T^{2} - 54 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 74 T + 34767 T^{2} + 74 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 78 T - 75761 T^{2} - 78 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 1106 T + 961995 T^{2} + 1106 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 492 T + 991654 T^{2} - 492 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 1640 T + 2303382 T^{2} + 1640 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 5538 T + 20405047 T^{2} - 5538 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3894 T + 41491891 T^{2} - 3894 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4718 T + 21950583 T^{2} + 4718 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4796 T - 2933298 T^{2} + 4796 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 15354 T + 290606395 T^{2} + 15354 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 32858 T + 548530503 T^{2} + 32858 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 24954 T + 600973327 T^{2} + 24954 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 16332 T + 896230798 T^{2} + 16332 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 21966 T + 1529583151 T^{2} + 21966 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 50 p T + 1583245203 T^{2} + 50 p^{6} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 36758 T + 36341997 p T^{2} + 36758 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 73848 T + 4273554814 T^{2} + 73848 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 51188 T + 63530214 p T^{2} + 51188 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 14926 T + 6101841783 T^{2} - 14926 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 90762 T + 9932848903 T^{2} + 90762 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 9300 T + 3189231862 T^{2} + 9300 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 30262 T + 8068652859 T^{2} - 30262 p^{5} T^{3} + p^{10} T^{4} \) |
| 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
−11.61298425393125072477097267809, −11.11972086089884166513777481985, −10.26092463063207030354359232201, −10.18230866603900699321846591041, −9.591474388889140797939902171336, −9.442414125922546825429919912384, −8.540636407442362719359062707768, −8.402470482006938691222235214975, −7.37433833507661145708285367591, −7.13806529114169311531517972716, −6.46618867659138973536813763792, −6.10113001626321159582088030157, −5.09426717300798439183170142182, −4.78352265788203075760844385039, −3.21747633950733596213687698259, −2.97438701421957663850008540384, −1.77957628053872271084307267445, −1.62572664444521784498626883956, 0, 0,
1.62572664444521784498626883956, 1.77957628053872271084307267445, 2.97438701421957663850008540384, 3.21747633950733596213687698259, 4.78352265788203075760844385039, 5.09426717300798439183170142182, 6.10113001626321159582088030157, 6.46618867659138973536813763792, 7.13806529114169311531517972716, 7.37433833507661145708285367591, 8.402470482006938691222235214975, 8.540636407442362719359062707768, 9.442414125922546825429919912384, 9.591474388889140797939902171336, 10.18230866603900699321846591041, 10.26092463063207030354359232201, 11.11972086089884166513777481985, 11.61298425393125072477097267809
