| L(s) = 1 | + 32·2-s + 108·3-s + 640·4-s − 328·5-s + 3.45e3·6-s + 1.37e3·7-s + 1.02e4·8-s + 7.29e3·9-s − 1.04e4·10-s − 7.94e3·11-s + 6.91e4·12-s − 8.78e3·13-s + 4.39e4·14-s − 3.54e4·15-s + 1.43e5·16-s − 2.72e4·17-s + 2.33e5·18-s − 3.48e4·19-s − 2.09e5·20-s + 1.48e5·21-s − 2.54e5·22-s − 2.23e4·23-s + 1.10e6·24-s − 1.34e5·25-s − 2.81e5·26-s + 3.93e5·27-s + 8.78e5·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 2.30·3-s + 5·4-s − 1.17·5-s + 6.53·6-s + 1.51·7-s + 7.07·8-s + 10/3·9-s − 3.31·10-s − 1.79·11-s + 11.5·12-s − 1.10·13-s + 4.27·14-s − 2.71·15-s + 35/4·16-s − 1.34·17-s + 9.42·18-s − 1.16·19-s − 5.86·20-s + 3.49·21-s − 5.09·22-s − 0.382·23-s + 16.3·24-s − 1.72·25-s − 3.13·26-s + 3.84·27-s + 7.55·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| 3 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| 7 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| 13 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 328 T + 242367 T^{2} + 14672484 p T^{3} + 1029550684 p^{2} T^{4} + 14672484 p^{8} T^{5} + 242367 p^{14} T^{6} + 328 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 7945 T + 91875558 T^{2} + 461778521445 T^{3} + 2812719259417354 T^{4} + 461778521445 p^{7} T^{5} + 91875558 p^{14} T^{6} + 7945 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 1605 p T + 850177728 T^{2} + 23433982507971 T^{3} + 483364201753843262 T^{4} + 23433982507971 p^{7} T^{5} + 850177728 p^{14} T^{6} + 1605 p^{22} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 34802 T + 3545801705 T^{2} + 87065010954882 T^{3} + 4755706410761676788 T^{4} + 87065010954882 p^{7} T^{5} + 3545801705 p^{14} T^{6} + 34802 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 22316 T + 2920860507 T^{2} - 149514667862280 T^{3} + 2298876745672108000 T^{4} - 149514667862280 p^{7} T^{5} + 2920860507 p^{14} T^{6} + 22316 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 158470 T + 32001781833 T^{2} + 5090062186428222 T^{3} + \)\(57\!\cdots\!32\)\( T^{4} + 5090062186428222 p^{7} T^{5} + 32001781833 p^{14} T^{6} + 158470 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 274647 T + 102551724716 T^{2} + 20246960617629963 T^{3} + \)\(41\!\cdots\!46\)\( T^{4} + 20246960617629963 p^{7} T^{5} + 102551724716 p^{14} T^{6} + 274647 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 319169 T + 238715551300 T^{2} + 58247796169084811 T^{3} + \)\(31\!\cdots\!22\)\( T^{4} + 58247796169084811 p^{7} T^{5} + 238715551300 p^{14} T^{6} + 319169 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 711254 T + 779328648444 T^{2} + 372857792853738114 T^{3} + \)\(22\!\cdots\!06\)\( T^{4} + 372857792853738114 p^{7} T^{5} + 779328648444 p^{14} T^{6} + 711254 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 893034 T + 1176199398229 T^{2} + 639239783919697818 T^{3} + \)\(47\!\cdots\!08\)\( T^{4} + 639239783919697818 p^{7} T^{5} + 1176199398229 p^{14} T^{6} + 893034 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 563129 T + 1446464751060 T^{2} + 631101111076468089 T^{3} + \)\(97\!\cdots\!26\)\( T^{4} + 631101111076468089 p^{7} T^{5} + 1446464751060 p^{14} T^{6} + 563129 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 2186067 T + 5140403302050 T^{2} + 6281553460320757257 T^{3} + \)\(86\!\cdots\!62\)\( T^{4} + 6281553460320757257 p^{7} T^{5} + 5140403302050 p^{14} T^{6} + 2186067 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 53200 T - 1885446746880 T^{2} + 1288598497313990208 T^{3} + \)\(94\!\cdots\!10\)\( T^{4} + 1288598497313990208 p^{7} T^{5} - 1885446746880 p^{14} T^{6} - 53200 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 2419497 T + 2462625203714 T^{2} - 9126731508135010125 T^{3} - \)\(22\!\cdots\!34\)\( T^{4} - 9126731508135010125 p^{7} T^{5} + 2462625203714 p^{14} T^{6} + 2419497 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 4908916 T + 25552994350336 T^{2} + 70622565025568368468 T^{3} + \)\(21\!\cdots\!58\)\( T^{4} + 70622565025568368468 p^{7} T^{5} + 25552994350336 p^{14} T^{6} + 4908916 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 5546448 T + 20460696752336 T^{2} + 34548406824922809264 T^{3} + \)\(80\!\cdots\!90\)\( T^{4} + 34548406824922809264 p^{7} T^{5} + 20460696752336 p^{14} T^{6} + 5546448 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 6114536 T + 31708929426395 T^{2} - \)\(13\!\cdots\!04\)\( T^{3} + \)\(52\!\cdots\!12\)\( T^{4} - \)\(13\!\cdots\!04\)\( p^{7} T^{5} + 31708929426395 p^{14} T^{6} - 6114536 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 6523461 T + 22956160781870 T^{2} + 34327557124976397423 T^{3} - \)\(39\!\cdots\!02\)\( T^{4} + 34327557124976397423 p^{7} T^{5} + 22956160781870 p^{14} T^{6} - 6523461 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 5265065 T + 85467572178570 T^{2} - \)\(33\!\cdots\!05\)\( T^{3} + \)\(32\!\cdots\!58\)\( T^{4} - \)\(33\!\cdots\!05\)\( p^{7} T^{5} + 85467572178570 p^{14} T^{6} - 5265065 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 4679151 T + 146629313208438 T^{2} + \)\(50\!\cdots\!21\)\( T^{3} + \)\(91\!\cdots\!14\)\( T^{4} + \)\(50\!\cdots\!21\)\( p^{7} T^{5} + 146629313208438 p^{14} T^{6} + 4679151 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 12566137 T + 179720231630510 T^{2} - \)\(60\!\cdots\!95\)\( T^{3} + \)\(85\!\cdots\!74\)\( T^{4} - \)\(60\!\cdots\!95\)\( p^{7} T^{5} + 179720231630510 p^{14} T^{6} - 12566137 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−7.26259890694352644435136102816, −6.65576175552575709822157157806, −6.64781676366074957363477366502, −6.38337535251010365409603774691, −6.35249971846430447711363488556, −5.45316071589083401357065010514, −5.42434200884045332486918360740, −5.27091528430559779852272333158, −5.26674390957929513998988647240, −4.59687915556786843865009568906, −4.53618851491565665280085027304, −4.44645071339704296204032260652, −4.28231602135126732879219447223, −3.66868615758597697898631458301, −3.56665870064871692487906485393, −3.52745563909564851679608059800, −3.46083514171306995263320742644, −2.72220301391384497041255264239, −2.50560858660904269749806690551, −2.43428744318733589074823890131, −2.35961904010077971197813594639, −1.75241698721786505969366694300, −1.64556182205884265481490612431, −1.58655683938537044913434731101, −1.35166618176504357570163949549, 0, 0, 0, 0,
1.35166618176504357570163949549, 1.58655683938537044913434731101, 1.64556182205884265481490612431, 1.75241698721786505969366694300, 2.35961904010077971197813594639, 2.43428744318733589074823890131, 2.50560858660904269749806690551, 2.72220301391384497041255264239, 3.46083514171306995263320742644, 3.52745563909564851679608059800, 3.56665870064871692487906485393, 3.66868615758597697898631458301, 4.28231602135126732879219447223, 4.44645071339704296204032260652, 4.53618851491565665280085027304, 4.59687915556786843865009568906, 5.26674390957929513998988647240, 5.27091528430559779852272333158, 5.42434200884045332486918360740, 5.45316071589083401357065010514, 6.35249971846430447711363488556, 6.38337535251010365409603774691, 6.64781676366074957363477366502, 6.65576175552575709822157157806, 7.26259890694352644435136102816
