| L(s) = 1 | − 23·3-s + 500·5-s + 1.37e3·7-s − 818·9-s − 1.84e3·11-s − 4.36e3·13-s − 1.15e4·15-s − 4.29e3·17-s − 5.39e4·19-s − 3.15e4·21-s − 4.61e3·23-s + 1.56e5·25-s − 2.47e4·27-s + 1.89e5·29-s − 2.91e5·31-s + 4.23e4·33-s + 6.86e5·35-s − 4.97e5·37-s + 1.00e5·39-s − 4.32e4·41-s − 5.88e5·43-s − 4.09e5·45-s − 1.30e6·47-s + 1.17e6·49-s + 9.86e4·51-s + 1.40e6·53-s − 9.20e5·55-s + ⋯ |
| L(s) = 1 | − 0.491·3-s + 1.78·5-s + 1.51·7-s − 0.374·9-s − 0.417·11-s − 0.551·13-s − 0.879·15-s − 0.211·17-s − 1.80·19-s − 0.743·21-s − 0.0790·23-s + 2·25-s − 0.242·27-s + 1.44·29-s − 1.75·31-s + 0.205·33-s + 2.70·35-s − 1.61·37-s + 0.271·39-s − 0.0981·41-s − 1.12·43-s − 0.669·45-s − 1.83·47-s + 10/7·49-s + 0.104·51-s + 1.29·53-s − 0.746·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| 7 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| good | 3 | $C_2 \wr S_4$ | \( 1 + 23 T + 449 p T^{2} + 24860 p T^{3} + 91340 p^{2} T^{4} + 24860 p^{8} T^{5} + 449 p^{15} T^{6} + 23 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 1841 T + 5219113 p T^{2} + 77146300468 T^{3} + 1510188850223324 T^{4} + 77146300468 p^{7} T^{5} + 5219113 p^{15} T^{6} + 1841 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 4369 T + 11566889 p T^{2} + 1027128532338 T^{3} + 11160896140216898 T^{4} + 1027128532338 p^{7} T^{5} + 11566889 p^{15} T^{6} + 4369 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 4291 T + 716757133 T^{2} - 209806868470 T^{3} + 256891910426522330 T^{4} - 209806868470 p^{7} T^{5} + 716757133 p^{14} T^{6} + 4291 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 53986 T + 2226065584 T^{2} + 91565862083706 T^{3} + 2979880905896829006 T^{4} + 91565862083706 p^{7} T^{5} + 2226065584 p^{14} T^{6} + 53986 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 4614 T + 5878146856 T^{2} + 203217650298150 T^{3} + 16337705859810052622 T^{4} + 203217650298150 p^{7} T^{5} + 5878146856 p^{14} T^{6} + 4614 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 189507 T + 33915652117 T^{2} - 3311117478074862 T^{3} + \)\(41\!\cdots\!82\)\( T^{4} - 3311117478074862 p^{7} T^{5} + 33915652117 p^{14} T^{6} - 189507 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 291584 T + 123846442188 T^{2} + 23503317503478016 T^{3} + \)\(52\!\cdots\!42\)\( T^{4} + 23503317503478016 p^{7} T^{5} + 123846442188 p^{14} T^{6} + 291584 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 497160 T + 360339474076 T^{2} + 123276810765790360 T^{3} + \)\(49\!\cdots\!38\)\( T^{4} + 123276810765790360 p^{7} T^{5} + 360339474076 p^{14} T^{6} + 497160 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 43294 T + 182375367748 T^{2} + 79478583423270698 T^{3} + \)\(48\!\cdots\!54\)\( T^{4} + 79478583423270698 p^{7} T^{5} + 182375367748 p^{14} T^{6} + 43294 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 588978 T + 680357697632 T^{2} + 348959579941841850 T^{3} + \)\(21\!\cdots\!10\)\( T^{4} + 348959579941841850 p^{7} T^{5} + 680357697632 p^{14} T^{6} + 588978 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 1302719 T + 1174764961495 T^{2} + 536705380579473232 T^{3} + \)\(31\!\cdots\!28\)\( T^{4} + 536705380579473232 p^{7} T^{5} + 1174764961495 p^{14} T^{6} + 1302719 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 1401046 T + 3902571899012 T^{2} - 3978983941386203386 T^{3} + \)\(62\!\cdots\!50\)\( T^{4} - 3978983941386203386 p^{7} T^{5} + 3902571899012 p^{14} T^{6} - 1401046 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 5299440 T + 16601089390124 T^{2} + 35866572245209181296 T^{3} + \)\(62\!\cdots\!30\)\( T^{4} + 35866572245209181296 p^{7} T^{5} + 16601089390124 p^{14} T^{6} + 5299440 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 590254 T + 5507156590260 T^{2} - 5389022606473826546 T^{3} + \)\(23\!\cdots\!98\)\( T^{4} - 5389022606473826546 p^{7} T^{5} + 5507156590260 p^{14} T^{6} - 590254 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 2050188 T + 12837436732780 T^{2} + 24848510556377976588 T^{3} + \)\(80\!\cdots\!62\)\( T^{4} + 24848510556377976588 p^{7} T^{5} + 12837436732780 p^{14} T^{6} + 2050188 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 5093280 T + 39426977453660 T^{2} + \)\(13\!\cdots\!60\)\( T^{3} + \)\(55\!\cdots\!58\)\( T^{4} + \)\(13\!\cdots\!60\)\( p^{7} T^{5} + 39426977453660 p^{14} T^{6} + 5093280 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 3421844 T + 23410102207764 T^{2} - 16466391352778085388 T^{3} + \)\(17\!\cdots\!18\)\( T^{4} - 16466391352778085388 p^{7} T^{5} + 23410102207764 p^{14} T^{6} - 3421844 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 6805877 T + 66929007859711 T^{2} - \)\(29\!\cdots\!68\)\( T^{3} + \)\(17\!\cdots\!76\)\( T^{4} - \)\(29\!\cdots\!68\)\( p^{7} T^{5} + 66929007859711 p^{14} T^{6} - 6805877 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 23010672 T + 279559830366092 T^{2} + \)\(22\!\cdots\!40\)\( T^{3} + \)\(13\!\cdots\!70\)\( T^{4} + \)\(22\!\cdots\!40\)\( p^{7} T^{5} + 279559830366092 p^{14} T^{6} + 23010672 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 12009486 T + 140729963022732 T^{2} - \)\(11\!\cdots\!18\)\( T^{3} + \)\(90\!\cdots\!34\)\( T^{4} - \)\(11\!\cdots\!18\)\( p^{7} T^{5} + 140729963022732 p^{14} T^{6} - 12009486 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 15784097 T + 348141161049757 T^{2} - \)\(35\!\cdots\!06\)\( T^{3} + \)\(42\!\cdots\!58\)\( T^{4} - \)\(35\!\cdots\!06\)\( p^{7} T^{5} + 348141161049757 p^{14} T^{6} - 15784097 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−6.86415839470892690296474065945, −6.79312539020334780521691927868, −6.53927737651158210896437858417, −6.39108720520702257373911097343, −6.14152072471087916161031701613, −5.64293962162803919313203956656, −5.60819534026104751746504406624, −5.57138506939777479204377563755, −5.23516041344193878636610488777, −4.78188334770872233625924141598, −4.73845631095267477141280713997, −4.51271795490012306214709342721, −4.48679056932841063043059023368, −3.89469098968383176601997041193, −3.44049499500489645501761430118, −3.37298742436577098688549836655, −3.11126533792157646363288353553, −2.47028650990758577298880963268, −2.36389867406367592156463991864, −2.30709632645013770790515273156, −1.90045831431348955535838036878, −1.68987652923591228909114626355, −1.38488463509082420785888413413, −1.11816606103082075197554105004, −1.08087846237393215586769039860, 0, 0, 0, 0,
1.08087846237393215586769039860, 1.11816606103082075197554105004, 1.38488463509082420785888413413, 1.68987652923591228909114626355, 1.90045831431348955535838036878, 2.30709632645013770790515273156, 2.36389867406367592156463991864, 2.47028650990758577298880963268, 3.11126533792157646363288353553, 3.37298742436577098688549836655, 3.44049499500489645501761430118, 3.89469098968383176601997041193, 4.48679056932841063043059023368, 4.51271795490012306214709342721, 4.73845631095267477141280713997, 4.78188334770872233625924141598, 5.23516041344193878636610488777, 5.57138506939777479204377563755, 5.60819534026104751746504406624, 5.64293962162803919313203956656, 6.14152072471087916161031701613, 6.39108720520702257373911097343, 6.53927737651158210896437858417, 6.79312539020334780521691927868, 6.86415839470892690296474065945
