| L(s) = 1 | + 8·2-s + 48·4-s − 108·5-s + 256·8-s − 864·10-s − 124·11-s + 720·13-s + 1.28e3·16-s + 1.26e3·17-s + 360·19-s − 5.18e3·20-s − 992·22-s − 6.52e3·23-s + 4.94e3·25-s + 5.76e3·26-s − 7.08e3·29-s + 5.90e3·31-s + 6.14e3·32-s + 1.00e4·34-s − 6.04e3·37-s + 2.88e3·38-s − 2.76e4·40-s + 1.73e4·41-s − 608·43-s − 5.95e3·44-s − 5.21e4·46-s + 3.04e4·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 1.93·5-s + 1.41·8-s − 2.73·10-s − 0.308·11-s + 1.18·13-s + 5/4·16-s + 1.05·17-s + 0.228·19-s − 2.89·20-s − 0.436·22-s − 2.57·23-s + 1.58·25-s + 1.67·26-s − 1.56·29-s + 1.10·31-s + 1.06·32-s + 1.49·34-s − 0.725·37-s + 0.323·38-s − 2.73·40-s + 1.61·41-s − 0.0501·43-s − 0.463·44-s − 3.63·46-s + 2.01·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 108 T + 6716 T^{2} + 108 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 124 T + 294194 T^{2} + 124 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 720 T + 673736 T^{2} - 720 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 1260 T + 2796692 T^{2} - 1260 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 360 T + 4777230 T^{2} - 360 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6524 T + 1009894 p T^{2} + 6524 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 7088 T + 48216146 T^{2} + 7088 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 5904 T + 58828406 T^{2} - 5904 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6040 T + 111103002 T^{2} + 6040 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 17388 T + 216792980 T^{2} - 17388 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 608 T + 164053110 T^{2} + 608 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 648 p T + 629420198 T^{2} - 648 p^{6} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3964 T + 594431822 T^{2} + 3964 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 40752 T + 1841031182 T^{2} + 40752 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1368 T + 1545466296 T^{2} + 1368 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 16224 T + 813179750 T^{2} + 16224 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3204 T - 212201462 T^{2} - 3204 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 23976 T + 68704368 T^{2} - 23976 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1040 p T + 7785668670 T^{2} + 1040 p^{6} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 173736 T + 14732805782 T^{2} + 173736 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 200556 T + 21158382260 T^{2} + 200556 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 251928 T + 32348722272 T^{2} - 251928 p^{5} T^{3} + p^{10} T^{4} \) |
| 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
−9.047409172072625776713966181973, −8.590615522744635089646514918710, −8.016107556012015818949064595035, −7.88277245126493696555168659416, −7.38961933418289585681318955410, −7.28099661275107127614004906919, −6.31815670830155299166156541536, −6.14966986664153974324857214883, −5.54716310709169844178730456079, −5.36774037980630252284283547740, −4.37672752137482228456574187899, −4.20775747082601207721577958367, −3.74490234916446448722665794607, −3.68192868222775962197203958632, −2.81355463206960669057603142382, −2.51705136803645793995081085002, −1.47499430740729147525165745607, −1.19102157271984553626278081789, 0, 0,
1.19102157271984553626278081789, 1.47499430740729147525165745607, 2.51705136803645793995081085002, 2.81355463206960669057603142382, 3.68192868222775962197203958632, 3.74490234916446448722665794607, 4.20775747082601207721577958367, 4.37672752137482228456574187899, 5.36774037980630252284283547740, 5.54716310709169844178730456079, 6.14966986664153974324857214883, 6.31815670830155299166156541536, 7.28099661275107127614004906919, 7.38961933418289585681318955410, 7.88277245126493696555168659416, 8.016107556012015818949064595035, 8.590615522744635089646514918710, 9.047409172072625776713966181973
