| L(s) = 1 | + 55·3-s + 500·5-s − 1.37e3·7-s − 2.33e3·9-s + 633·11-s − 5.39e3·13-s + 2.75e4·15-s − 4.19e4·17-s + 4.19e4·19-s − 7.54e4·21-s + 6.36e4·23-s + 1.56e5·25-s − 1.89e5·27-s + 1.07e3·29-s − 7.97e4·31-s + 3.48e4·33-s − 6.86e5·35-s − 1.31e5·37-s − 2.96e5·39-s − 3.13e5·41-s − 6.04e5·43-s − 1.16e6·45-s − 2.42e3·47-s + 1.17e6·49-s − 2.30e6·51-s − 5.90e5·53-s + 3.16e5·55-s + ⋯ |
| L(s) = 1 | + 1.17·3-s + 1.78·5-s − 1.51·7-s − 1.06·9-s + 0.143·11-s − 0.681·13-s + 2.10·15-s − 2.07·17-s + 1.40·19-s − 1.77·21-s + 1.09·23-s + 2·25-s − 1.85·27-s + 0.00815·29-s − 0.480·31-s + 0.168·33-s − 2.70·35-s − 0.427·37-s − 0.801·39-s − 0.711·41-s − 1.15·43-s − 1.90·45-s − 0.00341·47-s + 10/7·49-s − 2.43·51-s − 0.545·53-s + 0.256·55-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 | $C_2 \wr S_4$ | \( 1 - 55 T + 595 p^{2} T^{2} - 25876 p^{2} T^{3} + 638780 p^{3} T^{4} - 25876 p^{9} T^{5} + 595 p^{16} T^{6} - 55 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 633 T + 66706107 T^{2} - 34594652852 T^{3} + 1863176843142772 T^{4} - 34594652852 p^{7} T^{5} + 66706107 p^{14} T^{6} - 633 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 5397 T + 156092165 T^{2} + 727866079554 T^{3} + 11898283361486538 T^{4} + 727866079554 p^{7} T^{5} + 156092165 p^{14} T^{6} + 5397 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 41987 T + 1530490813 T^{2} + 44766750965754 T^{3} + 956445558728951450 T^{4} + 44766750965754 p^{7} T^{5} + 1530490813 p^{14} T^{6} + 41987 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 41958 T + 2845223008 T^{2} - 87525497209038 T^{3} + 3764704370323786094 T^{4} - 87525497209038 p^{7} T^{5} + 2845223008 p^{14} T^{6} - 41958 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 63650 T + 2871018072 T^{2} - 1672325721582 p T^{3} + 370184288107936002 p T^{4} - 1672325721582 p^{8} T^{5} + 2871018072 p^{14} T^{6} - 63650 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 1071 T + 53670001077 T^{2} - 296483890155022 T^{3} + \)\(12\!\cdots\!46\)\( T^{4} - 296483890155022 p^{7} T^{5} + 53670001077 p^{14} T^{6} - 1071 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 79700 T + 61430965932 T^{2} + 2169792504556740 T^{3} + \)\(17\!\cdots\!62\)\( T^{4} + 2169792504556740 p^{7} T^{5} + 61430965932 p^{14} T^{6} + 79700 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 131804 T + 349480138292 T^{2} + 35953196648822420 T^{3} + \)\(48\!\cdots\!18\)\( T^{4} + 35953196648822420 p^{7} T^{5} + 349480138292 p^{14} T^{6} + 131804 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 313834 T + 771662292396 T^{2} + 179814794529245806 T^{3} + \)\(22\!\cdots\!46\)\( T^{4} + 179814794529245806 p^{7} T^{5} + 771662292396 p^{14} T^{6} + 313834 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 604338 T + 1157890697008 T^{2} + 488261442041039706 T^{3} + \)\(48\!\cdots\!38\)\( T^{4} + 488261442041039706 p^{7} T^{5} + 1157890697008 p^{14} T^{6} + 604338 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 2429 T + 1503769569903 T^{2} + 82605292177181528 T^{3} + \)\(10\!\cdots\!84\)\( T^{4} + 82605292177181528 p^{7} T^{5} + 1503769569903 p^{14} T^{6} + 2429 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 590946 T + 2859814844852 T^{2} + 485885506517786766 T^{3} + \)\(37\!\cdots\!46\)\( T^{4} + 485885506517786766 p^{7} T^{5} + 2859814844852 p^{14} T^{6} + 590946 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 437296 T + 4368373194796 T^{2} - 1154586257196251728 T^{3} + \)\(10\!\cdots\!98\)\( T^{4} - 1154586257196251728 p^{7} T^{5} + 4368373194796 p^{14} T^{6} + 437296 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 369710 T + 3215884737260 T^{2} + 4288160748894425778 T^{3} + \)\(14\!\cdots\!18\)\( T^{4} + 4288160748894425778 p^{7} T^{5} + 3215884737260 p^{14} T^{6} + 369710 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 537812 T + 23517935889356 T^{2} + 9510167712903439700 T^{3} + \)\(21\!\cdots\!66\)\( T^{4} + 9510167712903439700 p^{7} T^{5} + 23517935889356 p^{14} T^{6} + 537812 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 2096808 T + 5246586125276 T^{2} + 24035158516232876472 T^{3} - \)\(64\!\cdots\!42\)\( T^{4} + 24035158516232876472 p^{7} T^{5} + 5246586125276 p^{14} T^{6} - 2096808 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 6033412 T + 50675866010756 T^{2} + \)\(18\!\cdots\!24\)\( T^{3} + \)\(85\!\cdots\!66\)\( T^{4} + \)\(18\!\cdots\!24\)\( p^{7} T^{5} + 50675866010756 p^{14} T^{6} + 6033412 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 5554681 T + 40917271437703 T^{2} + \)\(13\!\cdots\!64\)\( T^{3} + \)\(77\!\cdots\!16\)\( T^{4} + \)\(13\!\cdots\!64\)\( p^{7} T^{5} + 40917271437703 p^{14} T^{6} + 5554681 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 2234760 T + 101320946994604 T^{2} - \)\(16\!\cdots\!20\)\( T^{3} + \)\(40\!\cdots\!46\)\( T^{4} - \)\(16\!\cdots\!20\)\( p^{7} T^{5} + 101320946994604 p^{14} T^{6} - 2234760 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 5347574 T + 122843518616164 T^{2} + \)\(72\!\cdots\!14\)\( T^{3} + \)\(70\!\cdots\!66\)\( T^{4} + \)\(72\!\cdots\!14\)\( p^{7} T^{5} + 122843518616164 p^{14} T^{6} + 5347574 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 2106023 T + 186191803694125 T^{2} + \)\(15\!\cdots\!70\)\( T^{3} + \)\(19\!\cdots\!34\)\( T^{4} + \)\(15\!\cdots\!70\)\( p^{7} T^{5} + 186191803694125 p^{14} T^{6} + 2106023 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−6.97095185686918388992447870190, −6.71938852896825874841029486655, −6.51598685578356579279094148490, −6.50478032242843653134553477345, −6.31307911230838162083974687284, −5.66126261858711388688948617033, −5.57157572872021387783652051406, −5.54867971784017807514545619925, −5.41545955511517295303443777602, −4.79338962454811181210782931836, −4.74463748679666579076122063622, −4.43148457427436790031181887673, −4.15941402745396865076555083771, −3.53566035888158014579048108681, −3.45713181739012386908831343994, −3.27409733214605732491437844379, −3.05733723705278099114519366193, −2.65960874789890863100708205067, −2.50308610242838390916822807043, −2.35306563687558881976516874999, −2.27325242550091957924281334960, −1.57446936130503493323997440864, −1.50906198493332264353390211055, −1.06230889455270100555500956806, −1.03062539916739409767473923698, 0, 0, 0, 0,
1.03062539916739409767473923698, 1.06230889455270100555500956806, 1.50906198493332264353390211055, 1.57446936130503493323997440864, 2.27325242550091957924281334960, 2.35306563687558881976516874999, 2.50308610242838390916822807043, 2.65960874789890863100708205067, 3.05733723705278099114519366193, 3.27409733214605732491437844379, 3.45713181739012386908831343994, 3.53566035888158014579048108681, 4.15941402745396865076555083771, 4.43148457427436790031181887673, 4.74463748679666579076122063622, 4.79338962454811181210782931836, 5.41545955511517295303443777602, 5.54867971784017807514545619925, 5.57157572872021387783652051406, 5.66126261858711388688948617033, 6.31307911230838162083974687284, 6.50478032242843653134553477345, 6.51598685578356579279094148490, 6.71938852896825874841029486655, 6.97095185686918388992447870190
