| L(s) = 1 | + 48·5-s + 216·9-s + 240·13-s + 240·17-s − 1.01e3·25-s + 432·29-s + 2.80e3·37-s − 2.92e3·41-s + 1.03e4·45-s + 6.72e3·49-s + 1.77e3·53-s + 1.26e4·61-s + 1.15e4·65-s + 560·73-s + 2.26e4·81-s + 1.15e4·85-s − 2.29e4·89-s − 3.72e3·97-s + 4.44e4·101-s − 5.96e4·109-s − 9.07e3·113-s + 5.18e4·117-s + 6.46e4·121-s − 5.82e4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.91·5-s + 8/3·9-s + 1.42·13-s + 0.830·17-s − 1.62·25-s + 0.513·29-s + 2.04·37-s − 1.74·41-s + 5.11·45-s + 2.80·49-s + 0.632·53-s + 3.40·61-s + 2.72·65-s + 0.105·73-s + 3.44·81-s + 1.59·85-s − 2.90·89-s − 0.396·97-s + 4.35·101-s − 5.01·109-s − 0.710·113-s + 3.78·117-s + 4.41·121-s − 3.72·125-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(17.14887070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.14887070\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 8 p^{3} T^{2} + 24028 T^{4} - 141992 p^{2} T^{6} + 865126 p^{4} T^{8} - 141992 p^{10} T^{10} + 24028 p^{16} T^{12} - 8 p^{27} T^{14} + p^{32} T^{16} \) |
| 5 | \( ( 1 - 24 T + 1372 T^{2} - 42024 T^{3} + 1035142 T^{4} - 42024 p^{4} T^{5} + 1372 p^{8} T^{6} - 24 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 7 | \( 1 - 6728 T^{2} + 29560476 T^{4} - 97383797752 T^{6} + 248996961551942 T^{8} - 97383797752 p^{8} T^{10} + 29560476 p^{16} T^{12} - 6728 p^{24} T^{14} + p^{32} T^{16} \) |
| 11 | \( 1 - 64600 T^{2} + 2083873756 T^{4} - 45907087190632 T^{6} + 765101105580719686 T^{8} - 45907087190632 p^{8} T^{10} + 2083873756 p^{16} T^{12} - 64600 p^{24} T^{14} + p^{32} T^{16} \) |
| 13 | \( ( 1 - 120 T + 68124 T^{2} - 3870408 T^{3} + 2251470470 T^{4} - 3870408 p^{4} T^{5} + 68124 p^{8} T^{6} - 120 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 17 | \( ( 1 - 120 T + 168220 T^{2} - 16065864 T^{3} + 919560422 p T^{4} - 16065864 p^{4} T^{5} + 168220 p^{8} T^{6} - 120 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 19 | \( 1 - 550872 T^{2} + 164466729436 T^{4} - 33320633036248296 T^{6} + \)\(49\!\cdots\!46\)\( T^{8} - 33320633036248296 p^{8} T^{10} + 164466729436 p^{16} T^{12} - 550872 p^{24} T^{14} + p^{32} T^{16} \) |
| 23 | \( 1 - 1025096 T^{2} + 455029229724 T^{4} - 118004258074639864 T^{6} + 51883034777379169766 p^{2} T^{8} - 118004258074639864 p^{8} T^{10} + 455029229724 p^{16} T^{12} - 1025096 p^{24} T^{14} + p^{32} T^{16} \) |
| 29 | \( ( 1 - 216 T + 1846876 T^{2} - 423831144 T^{3} + 1803449019142 T^{4} - 423831144 p^{4} T^{5} + 1846876 p^{8} T^{6} - 216 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 31 | \( 1 - 2366472 T^{2} + 1444435485724 T^{4} + 1509721701480657864 T^{6} - \)\(28\!\cdots\!50\)\( T^{8} + 1509721701480657864 p^{8} T^{10} + 1444435485724 p^{16} T^{12} - 2366472 p^{24} T^{14} + p^{32} T^{16} \) |
| 37 | \( ( 1 - 1400 T + 6773148 T^{2} - 6215850184 T^{3} + 17798225143046 T^{4} - 6215850184 p^{4} T^{5} + 6773148 p^{8} T^{6} - 1400 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 41 | \( ( 1 + 1464 T + 3213340 T^{2} + 1762829064 T^{3} - 832430784314 T^{4} + 1762829064 p^{4} T^{5} + 3213340 p^{8} T^{6} + 1464 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 43 | \( 1 - 14239448 T^{2} + 102258927107292 T^{4} - \)\(53\!\cdots\!24\)\( T^{6} + \)\(21\!\cdots\!14\)\( T^{8} - \)\(53\!\cdots\!24\)\( p^{8} T^{10} + 102258927107292 p^{16} T^{12} - 14239448 p^{24} T^{14} + p^{32} T^{16} \) |
| 47 | \( 1 + 1700088 T^{2} + 22769502102556 T^{4} - \)\(10\!\cdots\!76\)\( T^{6} + \)\(23\!\cdots\!86\)\( T^{8} - \)\(10\!\cdots\!76\)\( p^{8} T^{10} + 22769502102556 p^{16} T^{12} + 1700088 p^{24} T^{14} + p^{32} T^{16} \) |
| 53 | \( ( 1 - 888 T + 17878684 T^{2} - 38865956040 T^{3} + 153671441281030 T^{4} - 38865956040 p^{4} T^{5} + 17878684 p^{8} T^{6} - 888 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 59 | \( 1 - 59081816 T^{2} + 1713928392943068 T^{4} - \)\(32\!\cdots\!48\)\( T^{6} + \)\(45\!\cdots\!06\)\( T^{8} - \)\(32\!\cdots\!48\)\( p^{8} T^{10} + 1713928392943068 p^{16} T^{12} - 59081816 p^{24} T^{14} + p^{32} T^{16} \) |
| 61 | \( ( 1 - 6328 T + 54475164 T^{2} - 210175360136 T^{3} + 1063754656519814 T^{4} - 210175360136 p^{4} T^{5} + 54475164 p^{8} T^{6} - 6328 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 67 | \( 1 - 74176088 T^{2} + 3133751785162972 T^{4} - \)\(92\!\cdots\!92\)\( T^{6} + \)\(20\!\cdots\!38\)\( T^{8} - \)\(92\!\cdots\!92\)\( p^{8} T^{10} + 3133751785162972 p^{16} T^{12} - 74176088 p^{24} T^{14} + p^{32} T^{16} \) |
| 71 | \( 1 - 116180040 T^{2} + 5397632108208796 T^{4} - \)\(19\!\cdots\!68\)\( p T^{6} + \)\(28\!\cdots\!78\)\( T^{8} - \)\(19\!\cdots\!68\)\( p^{9} T^{10} + 5397632108208796 p^{16} T^{12} - 116180040 p^{24} T^{14} + p^{32} T^{16} \) |
| 73 | \( ( 1 - 280 T + 72747100 T^{2} - 19192386472 T^{3} + 2931543571401286 T^{4} - 19192386472 p^{4} T^{5} + 72747100 p^{8} T^{6} - 280 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 79 | \( 1 - 111270664 T^{2} + 86221148754724 p T^{4} - \)\(27\!\cdots\!16\)\( T^{6} + \)\(10\!\cdots\!62\)\( T^{8} - \)\(27\!\cdots\!16\)\( p^{8} T^{10} + 86221148754724 p^{17} T^{12} - 111270664 p^{24} T^{14} + p^{32} T^{16} \) |
| 83 | \( 1 - 238640216 T^{2} + 29479046032067292 T^{4} - \)\(23\!\cdots\!92\)\( T^{6} + \)\(13\!\cdots\!54\)\( T^{8} - \)\(23\!\cdots\!92\)\( p^{8} T^{10} + 29479046032067292 p^{16} T^{12} - 238640216 p^{24} T^{14} + p^{32} T^{16} \) |
| 89 | \( ( 1 + 11496 T + 204574044 T^{2} + 1764445531224 T^{3} + 208411912385206 p T^{4} + 1764445531224 p^{4} T^{5} + 204574044 p^{8} T^{6} + 11496 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 97 | \( ( 1 + 1864 T + 295260828 T^{2} + 647805539576 T^{3} + 36499931947241798 T^{4} + 647805539576 p^{4} T^{5} + 295260828 p^{8} T^{6} + 1864 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
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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
−5.37219671491835483622344158329, −5.34920600801402458498167291425, −4.96546062030563007489332825267, −4.86925375971031032269412550189, −4.67475350686703071827348980254, −4.58581243699050082093068756403, −4.17050158978599183144591757648, −4.14929514275081297234023564525, −3.92056999575437876520277425544, −3.78884031721314120859374095778, −3.65194786614921951072403757045, −3.59971977022250947087668379893, −3.14566764092306428004828900175, −2.95296985088972041612251133922, −2.57707470582259338561721827577, −2.32235744456316149041552490980, −2.28686998478896560241873101043, −2.02438160819656456912069754430, −1.72185977882040235572769748051, −1.71215176399189424347356942980, −1.21639678299000302856236050950, −1.15734026345308052042423487561, −0.934187368127885406595401334354, −0.68804769711466102980485309064, −0.24540354999262541751768330095,
0.24540354999262541751768330095, 0.68804769711466102980485309064, 0.934187368127885406595401334354, 1.15734026345308052042423487561, 1.21639678299000302856236050950, 1.71215176399189424347356942980, 1.72185977882040235572769748051, 2.02438160819656456912069754430, 2.28686998478896560241873101043, 2.32235744456316149041552490980, 2.57707470582259338561721827577, 2.95296985088972041612251133922, 3.14566764092306428004828900175, 3.59971977022250947087668379893, 3.65194786614921951072403757045, 3.78884031721314120859374095778, 3.92056999575437876520277425544, 4.14929514275081297234023564525, 4.17050158978599183144591757648, 4.58581243699050082093068756403, 4.67475350686703071827348980254, 4.86925375971031032269412550189, 4.96546062030563007489332825267, 5.34920600801402458498167291425, 5.37219671491835483622344158329
Plot not available for L-functions of degree greater than 10.
