| L(s) = 1 | + 81·3-s + 308·5-s + 6·7-s + 4.37e3·9-s + 1.46e3·11-s + 3.07e3·13-s + 2.49e4·15-s − 5.12e3·17-s + 1.41e4·19-s + 486·21-s + 5.99e4·23-s − 5.39e4·25-s + 1.96e5·27-s + 7.19e4·29-s + 1.77e5·31-s + 1.18e5·33-s + 1.84e3·35-s − 1.84e5·37-s + 2.48e5·39-s + 6.57e5·41-s − 1.20e5·43-s + 1.34e6·45-s + 6.65e5·47-s − 6.32e5·49-s − 4.15e5·51-s − 1.11e5·53-s + 4.52e5·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.10·5-s + 0.00661·7-s + 2·9-s + 0.332·11-s + 0.387·13-s + 1.90·15-s − 0.253·17-s + 0.473·19-s + 0.0114·21-s + 1.02·23-s − 0.691·25-s + 1.92·27-s + 0.548·29-s + 1.06·31-s + 0.575·33-s + 0.00728·35-s − 0.597·37-s + 0.671·39-s + 1.48·41-s − 0.231·43-s + 2.20·45-s + 0.934·47-s − 0.768·49-s − 0.438·51-s − 0.103·53-s + 0.366·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56623104 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56623104 ^{s/2} \, \Gamma_{\C}(s+7/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(24.40124309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(24.40124309\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{3} T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 - 308 T + 29771 p T^{2} - 198568 p^{3} T^{3} + 29771 p^{8} T^{4} - 308 p^{14} T^{5} + p^{21} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 6 T + 633009 T^{2} + 755680484 T^{3} + 633009 p^{7} T^{4} - 6 p^{14} T^{5} + p^{21} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 1468 T + 45698297 T^{2} - 69782910888 T^{3} + 45698297 p^{7} T^{4} - 1468 p^{14} T^{5} + p^{21} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 3070 T + 173200603 T^{2} - 376979800532 T^{3} + 173200603 p^{7} T^{4} - 3070 p^{14} T^{5} + p^{21} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 5126 T + 544788319 T^{2} - 3522950006956 T^{3} + 544788319 p^{7} T^{4} + 5126 p^{14} T^{5} + p^{21} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 14168 T + 2629082929 T^{2} - 24420941363344 T^{3} + 2629082929 p^{7} T^{4} - 14168 p^{14} T^{5} + p^{21} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 59980 T + 9299110757 T^{2} - 401761187205480 T^{3} + 9299110757 p^{7} T^{4} - 59980 p^{14} T^{5} + p^{21} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 71984 T + 36220767423 T^{2} - 2812856294138336 T^{3} + 36220767423 p^{7} T^{4} - 71984 p^{14} T^{5} + p^{21} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 177470 T + 46090613769 T^{2} - 6716129169791980 T^{3} + 46090613769 p^{7} T^{4} - 177470 p^{14} T^{5} + p^{21} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 184014 T + 195049926675 T^{2} + 32182724235362708 T^{3} + 195049926675 p^{7} T^{4} + 184014 p^{14} T^{5} + p^{21} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 657250 T + 723825522999 T^{2} - 265662818647436860 T^{3} + 723825522999 p^{7} T^{4} - 657250 p^{14} T^{5} + p^{21} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 120544 T + 428576897001 T^{2} + 71835202741444928 T^{3} + 428576897001 p^{7} T^{4} + 120544 p^{14} T^{5} + p^{21} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 665132 T + 1511637250333 T^{2} - 670608977743109480 T^{3} + 1511637250333 p^{7} T^{4} - 665132 p^{14} T^{5} + p^{21} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 111888 T + 1040410113207 T^{2} + 1654039153643739552 T^{3} + 1040410113207 p^{7} T^{4} + 111888 p^{14} T^{5} + p^{21} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 1033308 T + 3277712028009 T^{2} + 6635505382456546024 T^{3} + 3277712028009 p^{7} T^{4} + 1033308 p^{14} T^{5} + p^{21} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 3584306 T + 12850891330187 T^{2} - 22903447315526735372 T^{3} + 12850891330187 p^{7} T^{4} - 3584306 p^{14} T^{5} + p^{21} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 2344900 T + 16286664954257 T^{2} + 26420648207241727384 T^{3} + 16286664954257 p^{7} T^{4} + 2344900 p^{14} T^{5} + p^{21} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 6311524 T + 32817320993493 T^{2} - \)\(10\!\cdots\!68\)\( T^{3} + 32817320993493 p^{7} T^{4} - 6311524 p^{14} T^{5} + p^{21} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 2112174 T + 10198609583895 T^{2} + 74299766409850042276 T^{3} + 10198609583895 p^{7} T^{4} + 2112174 p^{14} T^{5} + p^{21} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 2018830 T + 8233375166521 T^{2} + 61959879108292353460 T^{3} + 8233375166521 p^{7} T^{4} - 2018830 p^{14} T^{5} + p^{21} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 7430484 T + 60554604130017 T^{2} + \)\(29\!\cdots\!80\)\( T^{3} + 60554604130017 p^{7} T^{4} + 7430484 p^{14} T^{5} + p^{21} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 13300410 T + 84898111029255 T^{2} + \)\(40\!\cdots\!84\)\( T^{3} + 84898111029255 p^{7} T^{4} + 13300410 p^{14} T^{5} + p^{21} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 2960414 T + 183604417573135 T^{2} - \)\(25\!\cdots\!68\)\( T^{3} + 183604417573135 p^{7} T^{4} - 2960414 p^{14} T^{5} + p^{21} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.054399549502015862062298861073, −8.552395149026283792557522590738, −8.385315429569370886409263634222, −8.194617841807431745306104222557, −7.61621109905871055236890567292, −7.49302499642291105421949325044, −7.14227645127912153903585133569, −6.57559756402578429756935215742, −6.50122831582392207659706877143, −6.22946614992029472051936611038, −5.46705011172647851440252742724, −5.44971372224529095307817240858, −5.10116121464551122188651401782, −4.25457972044535799854869706055, −4.19422022753022250126235347002, −4.11088868249971280108505285318, −3.20789761637468074919809984634, −3.03786228667935104125396053449, −2.92571462874581579605513022456, −2.25089198770484563996430673505, −2.02080670209711173544830168389, −1.66334816451859577170085496987, −1.28689340606081124153514402503, −0.64254868682927879166074100725, −0.63709261949317737701380822050,
0.63709261949317737701380822050, 0.64254868682927879166074100725, 1.28689340606081124153514402503, 1.66334816451859577170085496987, 2.02080670209711173544830168389, 2.25089198770484563996430673505, 2.92571462874581579605513022456, 3.03786228667935104125396053449, 3.20789761637468074919809984634, 4.11088868249971280108505285318, 4.19422022753022250126235347002, 4.25457972044535799854869706055, 5.10116121464551122188651401782, 5.44971372224529095307817240858, 5.46705011172647851440252742724, 6.22946614992029472051936611038, 6.50122831582392207659706877143, 6.57559756402578429756935215742, 7.14227645127912153903585133569, 7.49302499642291105421949325044, 7.61621109905871055236890567292, 8.194617841807431745306104222557, 8.385315429569370886409263634222, 8.552395149026283792557522590738, 9.054399549502015862062298861073
