| L(s) = 1 | − 4·2-s + 3·3-s + 4·4-s + 6·5-s − 12·6-s − 7-s + 2·8-s + 3·9-s − 24·10-s + 11-s + 12·12-s + 8·13-s + 4·14-s + 18·15-s − 16-s + 12·17-s − 12·18-s − 2·19-s + 24·20-s − 3·21-s − 4·22-s − 14·23-s + 6·24-s + 11·25-s − 32·26-s − 4·28-s − 9·29-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 1.73·3-s + 2·4-s + 2.68·5-s − 4.89·6-s − 0.377·7-s + 0.707·8-s + 9-s − 7.58·10-s + 0.301·11-s + 3.46·12-s + 2.21·13-s + 1.06·14-s + 4.64·15-s − 1/4·16-s + 2.91·17-s − 2.82·18-s − 0.458·19-s + 5.36·20-s − 0.654·21-s − 0.852·22-s − 2.91·23-s + 1.22·24-s + 11/5·25-s − 6.27·26-s − 0.755·28-s − 1.67·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3383591511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3383591511\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p T + 2 p T^{2} - p^{2} T^{3} + p^{2} T^{4} - p^{3} T^{5} + 2 p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 11 | \( 1 - T - 20 T^{2} + T^{3} + 309 T^{4} + p T^{5} - 20 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| good | 2 | \( ( 1 - T + p^{2} T^{2} - p T^{3} + 9 T^{4} - p^{2} T^{5} + p^{4} T^{6} - p^{3} T^{7} + p^{4} T^{8} )( 1 + 5 T + 13 T^{2} + 25 T^{3} + 39 T^{4} + 25 p T^{5} + 13 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 5 | \( 1 - 6 T + p^{2} T^{2} - 54 T^{3} + 84 T^{4} - 84 T^{5} + 79 p T^{6} - 1746 T^{7} + 5171 T^{8} - 1746 p T^{9} + 79 p^{3} T^{10} - 84 p^{3} T^{11} + 84 p^{4} T^{12} - 54 p^{5} T^{13} + p^{8} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( ( 1 - 4 T + 9 T^{2} - 8 T^{3} - 31 T^{4} - 8 p T^{5} + 9 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )( 1 + 5 T + 18 T^{2} + 55 T^{3} + 149 T^{4} + 55 p T^{5} + 18 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 13 | \( 1 - 8 T + p T^{2} + 80 T^{3} - 400 T^{4} + 1168 T^{5} - 2101 T^{6} - 18940 T^{7} + 139519 T^{8} - 18940 p T^{9} - 2101 p^{2} T^{10} + 1168 p^{3} T^{11} - 400 p^{4} T^{12} + 80 p^{5} T^{13} + p^{7} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 6 T + 19 T^{2} - 132 T^{3} + 829 T^{4} - 132 p T^{5} + 19 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + T - 18 T^{2} - 37 T^{3} + 305 T^{4} - 37 p T^{5} - 18 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 7 T - 8 T^{2} + 77 T^{3} + 1593 T^{4} + 77 p T^{5} - 8 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( 1 + 9 T + 74 T^{2} + 129 T^{3} - 54 T^{4} - 8388 T^{5} + 644 p T^{6} + 272682 T^{7} + 3330947 T^{8} + 272682 p T^{9} + 644 p^{3} T^{10} - 8388 p^{3} T^{11} - 54 p^{4} T^{12} + 129 p^{5} T^{13} + 74 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) |
| 31 | \( 1 + 3 T + 36 T^{2} - 23 T^{3} - 84 T^{4} - 5516 T^{5} - 776 p T^{6} - 63114 T^{7} - 303923 T^{8} - 63114 p T^{9} - 776 p^{3} T^{10} - 5516 p^{3} T^{11} - 84 p^{4} T^{12} - 23 p^{5} T^{13} + 36 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 37 | \( ( 1 - 12 T + 57 T^{2} - 440 T^{3} + 3921 T^{4} - 440 p T^{5} + 57 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 - 3 T - 59 T^{2} - 42 T^{3} + 1326 T^{4} + 9501 T^{5} + 17474 T^{6} - 345486 T^{7} - 294103 T^{8} - 345486 p T^{9} + 17474 p^{2} T^{10} + 9501 p^{3} T^{11} + 1326 p^{4} T^{12} - 42 p^{5} T^{13} - 59 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 43 | \( ( 1 - 3 T - 68 T^{2} + 27 T^{3} + 3693 T^{4} + 27 p T^{5} - 68 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 23 T + 327 T^{2} - 2796 T^{3} + 15428 T^{4} - 41679 T^{5} - 80266 T^{6} + 1119532 T^{7} - 8567157 T^{8} + 1119532 p T^{9} - 80266 p^{2} T^{10} - 41679 p^{3} T^{11} + 15428 p^{4} T^{12} - 2796 p^{5} T^{13} + 327 p^{6} T^{14} - 23 p^{7} T^{15} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 14 T + 43 T^{2} - 160 T^{3} + 2961 T^{4} - 160 p T^{5} + 43 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 3 T + 64 T^{2} + 51 T^{3} - 168 T^{4} + 20412 T^{5} - 3128 p T^{6} + 519906 T^{7} - 7825759 T^{8} + 519906 p T^{9} - 3128 p^{3} T^{10} + 20412 p^{3} T^{11} - 168 p^{4} T^{12} + 51 p^{5} T^{13} + 64 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( 1 - 29 T^{2} - 480 T^{3} - 2160 T^{4} + 21600 T^{5} + 186389 T^{6} - 128880 T^{7} - 11482801 T^{8} - 128880 p T^{9} + 186389 p^{2} T^{10} + 21600 p^{3} T^{11} - 2160 p^{4} T^{12} - 480 p^{5} T^{13} - 29 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( ( 1 + 6 T - 31 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 21 T + 95 T^{2} + 1371 T^{3} - 21536 T^{4} + 1371 p T^{5} + 95 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 4 T - 27 T^{2} + 500 T^{3} + 7301 T^{4} + 500 p T^{5} - 27 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( 1 + 22 T + 259 T^{2} + 146 T^{3} - 28378 T^{4} - 420788 T^{5} - 1482397 T^{6} + 19182626 T^{7} + 350855611 T^{8} + 19182626 p T^{9} - 1482397 p^{2} T^{10} - 420788 p^{3} T^{11} - 28378 p^{4} T^{12} + 146 p^{5} T^{13} + 259 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) |
| 83 | \( 1 + 17 T + 213 T^{2} + 3330 T^{3} + 42290 T^{4} + 490473 T^{5} + 4981814 T^{6} + 51032300 T^{7} + 525354219 T^{8} + 51032300 p T^{9} + 4981814 p^{2} T^{10} + 490473 p^{3} T^{11} + 42290 p^{4} T^{12} + 3330 p^{5} T^{13} + 213 p^{6} T^{14} + 17 p^{7} T^{15} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 18 T + 254 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( 1 - 6 T + p T^{2} - 1530 T^{3} + 9180 T^{4} + 36876 T^{5} + 3263 p T^{6} + 9403380 T^{7} - 144949561 T^{8} + 9403380 p T^{9} + 3263 p^{3} T^{10} + 36876 p^{3} T^{11} + 9180 p^{4} T^{12} - 1530 p^{5} T^{13} + p^{7} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.69300762539668280481065120313, −6.36981426161494575469330794501, −6.14866934119995292180428372767, −5.96203555161875450458734773773, −5.78262692902104318883977465513, −5.73600587681427051295515295388, −5.73359513960011081183564375796, −5.56325286848054776189836955105, −5.49419391252283575947147137615, −5.39467360116086427635480141781, −4.63612257304490819939521844948, −4.35635264462464371282617043427, −4.20861360920313053861576519602, −4.11193053311449633801127247473, −3.96803792175130361880542707432, −3.59883759348807982760420571243, −3.56882269449676017353387022836, −3.33698196508211097243001960173, −2.63884421669695572618177200637, −2.55966398499463621512819833532, −2.48485563390693407317802539316, −2.39534913888368332658669380092, −1.60727503260737922382578774019, −1.40426069247133108431275174212, −1.03052895502872335515758168140,
1.03052895502872335515758168140, 1.40426069247133108431275174212, 1.60727503260737922382578774019, 2.39534913888368332658669380092, 2.48485563390693407317802539316, 2.55966398499463621512819833532, 2.63884421669695572618177200637, 3.33698196508211097243001960173, 3.56882269449676017353387022836, 3.59883759348807982760420571243, 3.96803792175130361880542707432, 4.11193053311449633801127247473, 4.20861360920313053861576519602, 4.35635264462464371282617043427, 4.63612257304490819939521844948, 5.39467360116086427635480141781, 5.49419391252283575947147137615, 5.56325286848054776189836955105, 5.73359513960011081183564375796, 5.73600587681427051295515295388, 5.78262692902104318883977465513, 5.96203555161875450458734773773, 6.14866934119995292180428372767, 6.36981426161494575469330794501, 6.69300762539668280481065120313
Plot not available for L-functions of degree greater than 10.
