| L(s) = 1 | + 32·2-s + 108·3-s + 640·4-s − 500·5-s + 3.45e3·6-s + 559·7-s + 1.02e4·8-s + 7.29e3·9-s − 1.60e4·10-s + 4.96e3·11-s + 6.91e4·12-s − 8.78e3·13-s + 1.78e4·14-s − 5.40e4·15-s + 1.43e5·16-s + 2.11e3·17-s + 2.33e5·18-s + 1.90e4·19-s − 3.20e5·20-s + 6.03e4·21-s + 1.58e5·22-s + 7.01e4·23-s + 1.10e6·24-s + 1.56e5·25-s − 2.81e5·26-s + 3.93e5·27-s + 3.57e5·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 2.30·3-s + 5·4-s − 1.78·5-s + 6.53·6-s + 0.615·7-s + 7.07·8-s + 10/3·9-s − 5.05·10-s + 1.12·11-s + 11.5·12-s − 1.10·13-s + 1.74·14-s − 4.13·15-s + 35/4·16-s + 0.104·17-s + 9.42·18-s + 0.635·19-s − 8.94·20-s + 1.42·21-s + 3.17·22-s + 1.20·23-s + 16.3·24-s + 2·25-s − 3.13·26-s + 3.84·27-s + 3.07·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(383.7749995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(383.7749995\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| 5 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| 13 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| good | 7 | $C_2 \wr S_4$ | \( 1 - 559 T + 2071978 T^{2} - 660571283 T^{3} + 282231767942 p T^{4} - 660571283 p^{7} T^{5} + 2071978 p^{14} T^{6} - 559 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 4963 T + 35448824 T^{2} - 173980943003 T^{3} + 1126897715978942 T^{4} - 173980943003 p^{7} T^{5} + 35448824 p^{14} T^{6} - 4963 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 2113 T + 918494840 T^{2} - 2007700423039 T^{3} + 537543742304980078 T^{4} - 2007700423039 p^{7} T^{5} + 918494840 p^{14} T^{6} - 2113 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 19012 T + 1269620200 T^{2} - 25472850377268 T^{3} + 1942333963439829822 T^{4} - 25472850377268 p^{7} T^{5} + 1269620200 p^{14} T^{6} - 19012 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 70175 T + 4092728990 T^{2} - 63066293888115 T^{3} + 7592119767864557442 T^{4} - 63066293888115 p^{7} T^{5} + 4092728990 p^{14} T^{6} - 70175 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 259742 T + 49893917528 T^{2} - 5782003137000682 T^{3} + \)\(71\!\cdots\!14\)\( T^{4} - 5782003137000682 p^{7} T^{5} + 49893917528 p^{14} T^{6} - 259742 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 266062 T + 123200897740 T^{2} - 21222366551202774 T^{3} + \)\(52\!\cdots\!26\)\( T^{4} - 21222366551202774 p^{7} T^{5} + 123200897740 p^{14} T^{6} - 266062 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 165333 T + 177511099462 T^{2} - 35262701197451263 T^{3} + \)\(15\!\cdots\!22\)\( T^{4} - 35262701197451263 p^{7} T^{5} + 177511099462 p^{14} T^{6} - 165333 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 540595 T + 548151363530 T^{2} - 207454028401630373 T^{3} + \)\(14\!\cdots\!18\)\( T^{4} - 207454028401630373 p^{7} T^{5} + 548151363530 p^{14} T^{6} - 540595 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 343276 T + 472657154620 T^{2} - 329616708027514892 T^{3} + \)\(13\!\cdots\!18\)\( T^{4} - 329616708027514892 p^{7} T^{5} + 472657154620 p^{14} T^{6} - 343276 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 1341054 T + 2197227755708 T^{2} - 1825565240227478310 T^{3} + \)\(16\!\cdots\!46\)\( T^{4} - 1825565240227478310 p^{7} T^{5} + 2197227755708 p^{14} T^{6} - 1341054 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 796467 T + 4302290314754 T^{2} - 2729803046627265193 T^{3} + \)\(73\!\cdots\!38\)\( T^{4} - 2729803046627265193 p^{7} T^{5} + 4302290314754 p^{14} T^{6} - 796467 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 1803732 T + 4441464247724 T^{2} - 4896866325007865396 T^{3} + \)\(68\!\cdots\!90\)\( T^{4} - 4896866325007865396 p^{7} T^{5} + 4441464247724 p^{14} T^{6} - 1803732 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 763651 T + 8733260002666 T^{2} - 1256151248435954753 T^{3} + \)\(32\!\cdots\!98\)\( T^{4} - 1256151248435954753 p^{7} T^{5} + 8733260002666 p^{14} T^{6} - 763651 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 1775800 T + 2850884626540 T^{2} + 6485176851958969800 T^{3} - \)\(27\!\cdots\!06\)\( T^{4} + 6485176851958969800 p^{7} T^{5} + 2850884626540 p^{14} T^{6} - 1775800 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 6273543 T + 49269741085844 T^{2} - \)\(18\!\cdots\!43\)\( T^{3} + \)\(73\!\cdots\!22\)\( T^{4} - \)\(18\!\cdots\!43\)\( p^{7} T^{5} + 49269741085844 p^{14} T^{6} - 6273543 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 7106658 T + 60239912127712 T^{2} - \)\(24\!\cdots\!66\)\( T^{3} + \)\(10\!\cdots\!06\)\( T^{4} - \)\(24\!\cdots\!66\)\( p^{7} T^{5} + 60239912127712 p^{14} T^{6} - 7106658 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 9212539 T + 94191782097004 T^{2} - \)\(50\!\cdots\!67\)\( T^{3} + \)\(28\!\cdots\!82\)\( T^{4} - \)\(50\!\cdots\!67\)\( p^{7} T^{5} + 94191782097004 p^{14} T^{6} - 9212539 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 5765928 T + 65411041853420 T^{2} - \)\(13\!\cdots\!20\)\( T^{3} + \)\(15\!\cdots\!74\)\( T^{4} - \)\(13\!\cdots\!20\)\( p^{7} T^{5} + 65411041853420 p^{14} T^{6} - 5765928 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 7741385 T + 44821139374298 T^{2} - \)\(38\!\cdots\!75\)\( T^{3} + \)\(21\!\cdots\!14\)\( T^{4} - \)\(38\!\cdots\!75\)\( p^{7} T^{5} + 44821139374298 p^{14} T^{6} - 7741385 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 31942525 T + 606174773948368 T^{2} - \)\(80\!\cdots\!87\)\( T^{3} + \)\(82\!\cdots\!50\)\( T^{4} - \)\(80\!\cdots\!87\)\( p^{7} T^{5} + 606174773948368 p^{14} T^{6} - 31942525 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−7.15509454223733824662852138100, −6.52994646012311590723494410996, −6.40654794978078469384695945270, −6.30992522090021012605434824663, −6.29688088165582279655820161862, −5.23014007233202757726315655324, −5.22856503618940702475164586166, −5.04416414704483542171332582149, −4.90940759489050052717564330021, −4.36187976265539679891001281220, −4.35317100341412717800802517863, −4.07999918861826149713649157493, −4.00054956848067527853116061849, −3.43321727240327168975316257702, −3.29007671052958965355032315605, −3.16981078498485619259568628528, −3.08116114710429656910654326621, −2.40440817928327496838400941767, −2.31267574626909047854778660166, −2.09759250909962730108019993822, −2.02304187548441727318071124946, −1.05517082653900974879081451599, −0.944813222690140955981612360273, −0.838088673572243537017282129038, −0.69197433625817831991268156437,
0.69197433625817831991268156437, 0.838088673572243537017282129038, 0.944813222690140955981612360273, 1.05517082653900974879081451599, 2.02304187548441727318071124946, 2.09759250909962730108019993822, 2.31267574626909047854778660166, 2.40440817928327496838400941767, 3.08116114710429656910654326621, 3.16981078498485619259568628528, 3.29007671052958965355032315605, 3.43321727240327168975316257702, 4.00054956848067527853116061849, 4.07999918861826149713649157493, 4.35317100341412717800802517863, 4.36187976265539679891001281220, 4.90940759489050052717564330021, 5.04416414704483542171332582149, 5.22856503618940702475164586166, 5.23014007233202757726315655324, 6.29688088165582279655820161862, 6.30992522090021012605434824663, 6.40654794978078469384695945270, 6.52994646012311590723494410996, 7.15509454223733824662852138100
