| L(s) = 1 | + 54·3-s − 24·5-s + 686·7-s + 2.18e3·9-s − 2.12e3·11-s − 1.08e3·13-s − 1.29e3·15-s − 2.92e4·17-s + 2.58e4·19-s + 3.70e4·21-s − 6.83e4·23-s − 1.38e5·25-s + 7.87e4·27-s + 2.11e5·29-s − 4.35e5·31-s − 1.14e5·33-s − 1.64e4·35-s − 2.84e4·37-s − 5.85e4·39-s + 7.49e5·41-s − 3.97e5·43-s − 5.24e4·45-s − 8.40e5·47-s + 3.52e5·49-s − 1.57e6·51-s − 2.46e5·53-s + 5.09e4·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.0858·5-s + 0.755·7-s + 9-s − 0.481·11-s − 0.136·13-s − 0.0991·15-s − 1.44·17-s + 0.863·19-s + 0.872·21-s − 1.17·23-s − 1.77·25-s + 0.769·27-s + 1.60·29-s − 2.62·31-s − 0.555·33-s − 0.0649·35-s − 0.0922·37-s − 0.158·39-s + 1.69·41-s − 0.761·43-s − 0.0858·45-s − 1.18·47-s + 3/7·49-s − 1.66·51-s − 0.227·53-s + 0.0413·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 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
−9.920008020656749609368634890282, −9.663011799811670967490421113835, −9.056935480822121398349910696761, −8.809434974931487232808227402667, −8.081497639341731725676751717590, −7.896057158821520848723230235280, −7.42067919065733400505519419605, −7.03266049035976927905546805240, −6.10557969872377582538562061862, −5.90462715063349319626402174726, −4.85717634455822173801229733184, −4.76194510222791804775413367295, −3.86191499242577659594568848115, −3.69171602567160325832635592540, −2.71051504268510728633380481064, −2.40927697006983942398718360252, −1.64260146839291208649482400482, −1.41523886062137731365401721118, 0, 0,
1.41523886062137731365401721118, 1.64260146839291208649482400482, 2.40927697006983942398718360252, 2.71051504268510728633380481064, 3.69171602567160325832635592540, 3.86191499242577659594568848115, 4.76194510222791804775413367295, 4.85717634455822173801229733184, 5.90462715063349319626402174726, 6.10557969872377582538562061862, 7.03266049035976927905546805240, 7.42067919065733400505519419605, 7.896057158821520848723230235280, 8.081497639341731725676751717590, 8.809434974931487232808227402667, 9.056935480822121398349910696761, 9.663011799811670967490421113835, 9.920008020656749609368634890282
