| L(s) = 1 | + 16·4-s − 54·9-s + 324·11-s + 192·16-s − 624·23-s + 1.72e3·25-s + 2.72e3·29-s − 864·36-s − 2.79e3·37-s − 632·43-s + 5.18e3·44-s + 2.07e3·53-s + 2.04e3·64-s − 2.92e4·67-s − 9.69e3·71-s − 7.94e3·79-s + 2.18e3·81-s − 9.98e3·92-s − 1.74e4·99-s + 2.76e4·100-s − 2.10e3·107-s + 4.00e3·109-s + 2.31e4·113-s + 4.35e4·116-s + 7.62e3·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 4-s − 2/3·9-s + 2.67·11-s + 3/4·16-s − 1.17·23-s + 2.76·25-s + 3.23·29-s − 2/3·36-s − 2.03·37-s − 0.341·43-s + 2.67·44-s + 0.739·53-s + 1/2·64-s − 6.50·67-s − 1.92·71-s − 1.27·79-s + 1/3·81-s − 1.17·92-s − 1.78·99-s + 2.76·100-s − 0.183·107-s + 0.336·109-s + 1.81·113-s + 3.23·116-s + 0.520·121-s + 6.20e−5·127-s + 5.82e−5·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.976255749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.976255749\) |
| \(L(3)\) |
|
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 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 1726 T^{2} + 1491171 T^{4} - 1726 p^{8} T^{6} + p^{16} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 162 T + 35555 T^{2} - 162 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 50980 T^{2} + 1849726854 T^{4} - 50980 p^{8} T^{6} + p^{16} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 940 p^{2} T^{2} + 383334 p^{4} T^{4} - 940 p^{10} T^{6} + p^{16} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 432940 T^{2} + 79003273254 T^{4} - 432940 p^{8} T^{6} + p^{16} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 312 T - 165070 T^{2} + 312 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 1362 T + 1874795 T^{2} - 1362 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 941806 T^{2} + 1059591694899 T^{4} - 941806 p^{8} T^{6} + p^{16} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 1396 T + 4217094 T^{2} + 1396 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2317228 T^{2} + 5437495988070 T^{4} - 2317228 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 316 T - 3534234 T^{2} + 316 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 3578788 T^{2} - 6833831226042 T^{4} - 3578788 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 1038 T + 13256075 T^{2} - 1038 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 27657790 T^{2} + 481686647166339 T^{4} - 27657790 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 49777732 T^{2} + 997300062659526 T^{4} - 49777732 p^{8} T^{6} + p^{16} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 14600 T + 91702674 T^{2} + 14600 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 4848 T + 52412546 T^{2} + 4848 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 16586788 T^{2} + 661626455827398 T^{4} - 16586788 p^{8} T^{6} + p^{16} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 3974 T + 10279683 T^{2} + 3974 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1452506 p T^{2} + 7047387074675283 T^{4} - 1452506 p^{9} T^{6} + p^{16} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 3460244 T^{2} - 5170773883613466 T^{4} + 3460244 p^{8} T^{6} + p^{16} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 347883310 T^{2} + 45930093674217747 T^{4} - 347883310 p^{8} T^{6} + p^{16} T^{8} \) |
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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
−7.79733132420306646792413662315, −7.61103371748337570904146737444, −7.11668938432219116238751338687, −7.00831117513148691814014280988, −6.97181219342444501382787317064, −6.64551074602814481604158606473, −6.21007113766746081681559164631, −6.05304186361260594166247165473, −6.04153812385659127505052833517, −5.74171949192846465362162604864, −5.02279796169504370044407030169, −4.87357303413495353290725915332, −4.66096411192421070825510363734, −4.29709281645671827463454045155, −4.07042249533355394740469010646, −3.48803677532528364900591763919, −3.37965716463710990061763293407, −2.95566047227311979838443106410, −2.66090232162219306259740138736, −2.55170275761155799471976615417, −1.63829830168057256167751104237, −1.48552561992608855046474638427, −1.32127101339660347145523258816, −0.897683203720560599011824174760, −0.24419908145520704448089273093,
0.24419908145520704448089273093, 0.897683203720560599011824174760, 1.32127101339660347145523258816, 1.48552561992608855046474638427, 1.63829830168057256167751104237, 2.55170275761155799471976615417, 2.66090232162219306259740138736, 2.95566047227311979838443106410, 3.37965716463710990061763293407, 3.48803677532528364900591763919, 4.07042249533355394740469010646, 4.29709281645671827463454045155, 4.66096411192421070825510363734, 4.87357303413495353290725915332, 5.02279796169504370044407030169, 5.74171949192846465362162604864, 6.04153812385659127505052833517, 6.05304186361260594166247165473, 6.21007113766746081681559164631, 6.64551074602814481604158606473, 6.97181219342444501382787317064, 7.00831117513148691814014280988, 7.11668938432219116238751338687, 7.61103371748337570904146737444, 7.79733132420306646792413662315
