| L(s) = 1 | − 32·2-s − 108·3-s + 640·4-s − 500·5-s + 3.45e3·6-s + 1.00e3·7-s − 1.02e4·8-s + 7.29e3·9-s + 1.60e4·10-s − 4.51e3·11-s − 6.91e4·12-s + 8.78e3·13-s − 3.20e4·14-s + 5.40e4·15-s + 1.43e5·16-s − 2.61e4·17-s − 2.33e5·18-s − 3.50e3·19-s − 3.20e5·20-s − 1.08e5·21-s + 1.44e5·22-s − 9.19e4·23-s + 1.10e6·24-s + 1.56e5·25-s − 2.81e5·26-s − 3.93e5·27-s + 6.40e5·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 2.30·3-s + 5·4-s − 1.78·5-s + 6.53·6-s + 1.10·7-s − 7.07·8-s + 10/3·9-s + 5.05·10-s − 1.02·11-s − 11.5·12-s + 1.10·13-s − 3.11·14-s + 4.13·15-s + 35/4·16-s − 1.29·17-s − 9.42·18-s − 0.117·19-s − 8.94·20-s − 2.54·21-s + 2.89·22-s − 1.57·23-s + 16.3·24-s + 2·25-s − 3.13·26-s − 3.84·27-s + 5.51·28-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
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| 5 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| 13 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| good | 7 | $C_2 \wr S_4$ | \( 1 - 143 p T + 1432618 T^{2} - 53016891 p T^{3} + 774189623754 T^{4} - 53016891 p^{8} T^{5} + 1432618 p^{14} T^{6} - 143 p^{22} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 4517 T + 32932484 T^{2} + 85042183197 T^{3} + 731973009764742 T^{4} + 85042183197 p^{7} T^{5} + 32932484 p^{14} T^{6} + 4517 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 26153 T + 610291260 T^{2} - 5077071422721 T^{3} - 106707074404343402 T^{4} - 5077071422721 p^{7} T^{5} + 610291260 p^{14} T^{6} + 26153 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 3508 T + 1128328720 T^{2} + 18779745075092 T^{3} + 1323829000512582062 T^{4} + 18779745075092 p^{7} T^{5} + 1128328720 p^{14} T^{6} + 3508 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 91935 T + 15423898370 T^{2} + 951970472506315 T^{3} + 81876145422676191162 T^{4} + 951970472506315 p^{7} T^{5} + 15423898370 p^{14} T^{6} + 91935 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 98998 T + 35324301168 T^{2} + 4007699163597138 T^{3} + \)\(63\!\cdots\!34\)\( T^{4} + 4007699163597138 p^{7} T^{5} + 35324301168 p^{14} T^{6} + 98998 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 229342 T + 49361805700 T^{2} - 7288460481943814 T^{3} + \)\(10\!\cdots\!06\)\( T^{4} - 7288460481943814 p^{7} T^{5} + 49361805700 p^{14} T^{6} - 229342 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 87547 T + 301854775022 T^{2} - 14694603882636657 T^{3} + \)\(38\!\cdots\!62\)\( T^{4} - 14694603882636657 p^{7} T^{5} + 301854775022 p^{14} T^{6} - 87547 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 329365 T + 585370482990 T^{2} + 96403705005740787 T^{3} + \)\(14\!\cdots\!78\)\( T^{4} + 96403705005740787 p^{7} T^{5} + 585370482990 p^{14} T^{6} + 329365 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 79596 T + 688987970940 T^{2} + 120932330364675852 T^{3} + \)\(23\!\cdots\!18\)\( T^{4} + 120932330364675852 p^{7} T^{5} + 688987970940 p^{14} T^{6} + 79596 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 508546 T + 2083257882108 T^{2} - 773129341050686490 T^{3} + \)\(15\!\cdots\!46\)\( T^{4} - 773129341050686490 p^{7} T^{5} + 2083257882108 p^{14} T^{6} - 508546 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 375007 T + 3819614127674 T^{2} + 1296353293274479533 T^{3} + \)\(62\!\cdots\!98\)\( T^{4} + 1296353293274479533 p^{7} T^{5} + 3819614127674 p^{14} T^{6} + 375007 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 2286512 T + 5223939893404 T^{2} - 1873304913667794736 T^{3} + \)\(43\!\cdots\!50\)\( T^{4} - 1873304913667794736 p^{7} T^{5} + 5223939893404 p^{14} T^{6} - 2286512 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 2290091 T + 7853386289146 T^{2} - 11119222728849139513 T^{3} + \)\(25\!\cdots\!58\)\( T^{4} - 11119222728849139513 p^{7} T^{5} + 7853386289146 p^{14} T^{6} - 2290091 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 4214020 T + 19324565242380 T^{2} - 39745556806927734020 T^{3} + \)\(12\!\cdots\!94\)\( T^{4} - 39745556806927734020 p^{7} T^{5} + 19324565242380 p^{14} T^{6} - 4214020 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 5377363 T + 38895111544244 T^{2} - \)\(12\!\cdots\!63\)\( T^{3} + \)\(53\!\cdots\!82\)\( T^{4} - \)\(12\!\cdots\!63\)\( p^{7} T^{5} + 38895111544244 p^{14} T^{6} - 5377363 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 8629962 T + 47417948369792 T^{2} - \)\(21\!\cdots\!54\)\( T^{3} + \)\(82\!\cdots\!26\)\( T^{4} - \)\(21\!\cdots\!54\)\( p^{7} T^{5} + 47417948369792 p^{14} T^{6} - 8629962 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 8370699 T + 72988957733684 T^{2} - \)\(32\!\cdots\!27\)\( T^{3} + \)\(17\!\cdots\!42\)\( T^{4} - \)\(32\!\cdots\!27\)\( p^{7} T^{5} + 72988957733684 p^{14} T^{6} - 8370699 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 3863268 T + 24265882336220 T^{2} + 1076429892581198820 T^{3} - \)\(28\!\cdots\!06\)\( T^{4} + 1076429892581198820 p^{7} T^{5} + 24265882336220 p^{14} T^{6} + 3863268 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 261105 T + 156918459176798 T^{2} - 32671683685818658615 T^{3} + \)\(10\!\cdots\!54\)\( T^{4} - 32671683685818658615 p^{7} T^{5} + 156918459176798 p^{14} T^{6} - 261105 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 7664095 T + 236424698939508 T^{2} - \)\(15\!\cdots\!13\)\( T^{3} + \)\(26\!\cdots\!10\)\( T^{4} - \)\(15\!\cdots\!13\)\( p^{7} T^{5} + 236424698939508 p^{14} T^{6} - 7664095 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−7.80013348145226101584451228380, −7.22092357796347767279773201054, −7.05920566173561719486360466086, −6.83956235694975116987709032443, −6.52659739862865217132722712743, −6.44372349862324185521070489094, −6.12451196060936883940769062223, −5.92743997951137331151874716828, −5.67048174328199482800159202240, −5.09114951896482928572451756286, −4.94203010950450193707090808909, −4.86361498603739855181133408033, −4.68552982513169935280822298420, −3.83768242215847171868759927587, −3.79535008031279230682812911015, −3.70356808256117057993351121596, −3.47932666669767688746813089926, −2.55431809883223980755633351473, −2.21519903167681189357250763396, −2.19521301203451081156607738020, −2.15481320642024836470122008981, −1.14521344702658401443406757530, −1.12468673309469508044053833937, −1.07500261062143455940221711656, −0.822521309167342807367044896889, 0, 0, 0, 0,
0.822521309167342807367044896889, 1.07500261062143455940221711656, 1.12468673309469508044053833937, 1.14521344702658401443406757530, 2.15481320642024836470122008981, 2.19521301203451081156607738020, 2.21519903167681189357250763396, 2.55431809883223980755633351473, 3.47932666669767688746813089926, 3.70356808256117057993351121596, 3.79535008031279230682812911015, 3.83768242215847171868759927587, 4.68552982513169935280822298420, 4.86361498603739855181133408033, 4.94203010950450193707090808909, 5.09114951896482928572451756286, 5.67048174328199482800159202240, 5.92743997951137331151874716828, 6.12451196060936883940769062223, 6.44372349862324185521070489094, 6.52659739862865217132722712743, 6.83956235694975116987709032443, 7.05920566173561719486360466086, 7.22092357796347767279773201054, 7.80013348145226101584451228380
