| L(s) = 1 | − 12·2-s + 120·4-s + 24·5-s − 686·7-s − 2.68e3·8-s − 288·10-s − 2.12e3·11-s − 1.08e3·13-s + 8.23e3·14-s + 3.14e4·16-s + 2.92e4·17-s − 2.58e4·19-s + 2.88e3·20-s + 2.54e4·22-s − 6.83e4·23-s − 1.38e5·25-s + 1.30e4·26-s − 8.23e4·28-s − 2.11e5·29-s + 4.35e5·31-s − 3.00e5·32-s − 3.51e5·34-s − 1.64e4·35-s − 2.84e4·37-s + 3.09e5·38-s − 6.45e4·40-s − 7.49e5·41-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 0.937·4-s + 0.0858·5-s − 0.755·7-s − 1.85·8-s − 0.0910·10-s − 0.481·11-s − 0.136·13-s + 0.801·14-s + 1.91·16-s + 1.44·17-s − 0.863·19-s + 0.0804·20-s + 0.510·22-s − 1.17·23-s − 1.77·25-s + 0.145·26-s − 0.708·28-s − 1.60·29-s + 2.62·31-s − 1.61·32-s − 1.53·34-s − 0.0649·35-s − 0.0922·37-s + 0.915·38-s − 0.159·40-s − 1.69·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + 3 p^{2} T + 3 p^{3} T^{2} + 3 p^{9} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 24 T + 139242 T^{2} - 24 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2124 T + 19090986 T^{2} + 2124 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 1084 T + 103561806 T^{2} + 1084 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 29256 T + 533114098 T^{2} - 29256 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 25816 T + 1627717254 T^{2} + 25816 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 68316 T + 7976265490 T^{2} + 68316 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 211308 T + 35807598606 T^{2} + 211308 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 435840 T + 98660711870 T^{2} - 435840 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 28428 T + 188976571454 T^{2} + 28428 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 749760 T + 517210006962 T^{2} + 749760 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 397096 T + 472051884246 T^{2} - 397096 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 840168 T + 744642372910 T^{2} + 840168 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 246684 T + 2085795052990 T^{2} - 246684 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2199504 T + 4631611593574 T^{2} + 2199504 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1951108 T + 6790017439086 T^{2} + 1951108 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1532048 T + 530136191190 T^{2} - 1532048 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2024004 T + 14260375775986 T^{2} + 2024004 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1709028 T + 11306816102198 T^{2} + 1709028 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1048168 T + 11630316806382 T^{2} - 1048168 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4894296 T + 47206286808070 T^{2} - 4894296 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 60864 T + 81562257471570 T^{2} - 60864 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 26046852 T + 325711749428294 T^{2} + 26046852 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−13.26958312128209647494415304238, −12.20498866605396056222693933111, −12.18774040157728396274515498994, −11.63878141193753812554160723889, −10.69711145823181489123245581829, −10.16384891589984125128314142082, −9.553879987042199877743671748967, −9.503600216817222478984872729829, −8.220984722303518213485284237164, −8.198649941851839898673135066522, −7.35451683420467189464982656640, −6.35472362775751543602331120056, −6.09410619181540180490597491241, −5.35829811113176423122105055060, −3.96333725251308569480957695407, −3.17577243905850741820388652407, −2.43412819884927020429021780737, −1.42884040065161816048872546875, 0, 0,
1.42884040065161816048872546875, 2.43412819884927020429021780737, 3.17577243905850741820388652407, 3.96333725251308569480957695407, 5.35829811113176423122105055060, 6.09410619181540180490597491241, 6.35472362775751543602331120056, 7.35451683420467189464982656640, 8.198649941851839898673135066522, 8.220984722303518213485284237164, 9.503600216817222478984872729829, 9.553879987042199877743671748967, 10.16384891589984125128314142082, 10.69711145823181489123245581829, 11.63878141193753812554160723889, 12.18774040157728396274515498994, 12.20498866605396056222693933111, 13.26958312128209647494415304238
