| L(s) = 1 | + 2·2-s + 6·3-s + 9·4-s − 6·5-s + 12·6-s − 26·7-s + 30·8-s + 21·9-s − 12·10-s − 14·11-s + 54·12-s − 52·14-s − 36·15-s + 60·16-s − 12·17-s + 42·18-s − 36·19-s − 54·20-s − 156·21-s − 28·22-s + 24·23-s + 180·24-s + 23·25-s + 54·27-s − 234·28-s − 44·29-s − 72·30-s + ⋯ |
| L(s) = 1 | + 2-s + 2·3-s + 9/4·4-s − 6/5·5-s + 2·6-s − 3.71·7-s + 15/4·8-s + 7/3·9-s − 6/5·10-s − 1.27·11-s + 9/2·12-s − 3.71·14-s − 2.39·15-s + 15/4·16-s − 0.705·17-s + 7/3·18-s − 1.89·19-s − 2.69·20-s − 7.42·21-s − 1.27·22-s + 1.04·23-s + 15/2·24-s + 0.919·25-s + 2·27-s − 8.35·28-s − 1.51·29-s − 2.39·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.751436386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.751436386\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 13 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 2 | $C_2^2$ | \( ( 1 - T - 3 T^{2} - p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 + 6 T + 13 T^{2} + 6 T^{3} - 324 T^{4} + 6 p^{2} T^{5} + 13 p^{4} T^{6} + 6 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 14 T + 61 T^{2} - 1498 T^{3} - 20132 T^{4} - 1498 p^{2} T^{5} + 61 p^{4} T^{6} + 14 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 12 T + 586 T^{2} + 6456 T^{3} + 219795 T^{4} + 6456 p^{2} T^{5} + 586 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 36 T + 1054 T^{2} + 22392 T^{3} + 412515 T^{4} + 22392 p^{2} T^{5} + 1054 p^{4} T^{6} + 36 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 24 T - 470 T^{2} + 288 T^{3} + 509571 T^{4} + 288 p^{2} T^{5} - 470 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 22 T + 1647 T^{2} + 22 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 42 T + 79 p T^{2} - 78162 T^{3} + 3104868 T^{4} - 78162 p^{2} T^{5} + 79 p^{5} T^{6} - 42 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 76 T + 2218 T^{2} - 62320 T^{3} + 2877859 T^{4} - 62320 p^{2} T^{5} + 2218 p^{4} T^{6} - 76 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2956 T^{2} + 7746150 T^{4} - 2956 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 60 T + 4442 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 72 T + 6526 T^{2} + 345456 T^{3} + 21958275 T^{4} + 345456 p^{2} T^{5} + 6526 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 6 T - 5435 T^{2} - 882 T^{3} + 21940236 T^{4} - 882 p^{2} T^{5} - 5435 p^{4} T^{6} + 6 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 90 T + 10285 T^{2} + 682650 T^{3} + 54813564 T^{4} + 682650 p^{2} T^{5} + 10285 p^{4} T^{6} + 90 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 216 T + 24334 T^{2} + 1896912 T^{3} + 121146675 T^{4} + 1896912 p^{2} T^{5} + 24334 p^{4} T^{6} + 216 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 32 T + 4426 T^{2} - 396160 T^{3} - 13174253 T^{4} - 396160 p^{2} T^{5} + 4426 p^{4} T^{6} + 32 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 128 T + 13554 T^{2} + 128 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 144 T + 17426 T^{2} - 1514016 T^{3} + 118980003 T^{4} - 1514016 p^{2} T^{5} + 17426 p^{4} T^{6} - 144 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 58 T - 4343 T^{2} - 276950 T^{3} - 546956 T^{4} - 276950 p^{2} T^{5} - 4343 p^{4} T^{6} + 58 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 9974 T^{2} + 119681355 T^{4} - 9974 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 180 T + 26794 T^{2} - 2878920 T^{3} + 278612595 T^{4} - 2878920 p^{2} T^{5} + 26794 p^{4} T^{6} - 180 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 33742 T^{2} + 461127603 T^{4} - 33742 p^{4} T^{6} + p^{8} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.413651732990645340047478737460, −7.78121087797159997612550386925, −7.66759876971833657358184312084, −7.62589336987226050464123519893, −7.58973848380037574123523145559, −7.26600957489441227489781840137, −6.80795692485170425152750362394, −6.45899208670046344349423460273, −6.43244155628400655059680824363, −6.37412948498978196043496041113, −5.86828584202516141797694895866, −5.72234676394959471679115612703, −4.74158353550671570172949889772, −4.63911095989767290898248237243, −4.54764807850866370349303075319, −4.02004814224716778877081385894, −3.83101432917102071109868127445, −3.54980662167912210150031149924, −2.97243306967795448162206787942, −2.86094814637404049501993198712, −2.72801787140318122042311048046, −2.45437407280529728319383513320, −2.01592261676971746955323215762, −1.32570644620321659406179641156, −0.34482911443511969059224026445,
0.34482911443511969059224026445, 1.32570644620321659406179641156, 2.01592261676971746955323215762, 2.45437407280529728319383513320, 2.72801787140318122042311048046, 2.86094814637404049501993198712, 2.97243306967795448162206787942, 3.54980662167912210150031149924, 3.83101432917102071109868127445, 4.02004814224716778877081385894, 4.54764807850866370349303075319, 4.63911095989767290898248237243, 4.74158353550671570172949889772, 5.72234676394959471679115612703, 5.86828584202516141797694895866, 6.37412948498978196043496041113, 6.43244155628400655059680824363, 6.45899208670046344349423460273, 6.80795692485170425152750362394, 7.26600957489441227489781840137, 7.58973848380037574123523145559, 7.62589336987226050464123519893, 7.66759876971833657358184312084, 7.78121087797159997612550386925, 8.413651732990645340047478737460
