| L(s) = 1 | + 2·2-s + 4·4-s + 4·5-s + 2·7-s + 8·10-s − 4·11-s − 4·13-s + 4·14-s − 5·16-s + 2·17-s + 16·20-s − 8·22-s + 4·23-s + 16·25-s − 8·26-s + 8·28-s − 16·29-s + 12·31-s − 16·32-s + 4·34-s + 8·35-s + 4·37-s − 4·41-s + 36·43-s − 16·44-s + 8·46-s + 12·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 2·4-s + 1.78·5-s + 0.755·7-s + 2.52·10-s − 1.20·11-s − 1.10·13-s + 1.06·14-s − 5/4·16-s + 0.485·17-s + 3.57·20-s − 1.70·22-s + 0.834·23-s + 16/5·25-s − 1.56·26-s + 1.51·28-s − 2.97·29-s + 2.15·31-s − 2.82·32-s + 0.685·34-s + 1.35·35-s + 0.657·37-s − 0.624·41-s + 5.48·43-s − 2.41·44-s + 1.17·46-s + 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.243262800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.243262800\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - 2 T + 4 T^{2} - 10 T^{3} - 22 T^{4} - 10 p T^{5} + 4 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | \( ( 1 + T + T^{2} )^{4} \) |
| good | 2 | \( 1 - p T + p^{3} T^{3} - 11 T^{4} - p^{2} T^{5} + 5 p^{2} T^{6} - p^{3} T^{7} - 23 T^{8} - p^{4} T^{9} + 5 p^{4} T^{10} - p^{5} T^{11} - 11 p^{4} T^{12} + p^{8} T^{13} - p^{8} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 - 4 T + 16 T^{3} - 2 T^{4} - 56 T^{5} + 32 T^{6} + 76 p T^{7} - 1421 T^{8} + 76 p^{2} T^{9} + 32 p^{2} T^{10} - 56 p^{3} T^{11} - 2 p^{4} T^{12} + 16 p^{5} T^{13} - 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 13 | \( ( 1 + 2 T + 44 T^{2} + 62 T^{3} + 802 T^{4} + 62 p T^{5} + 44 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 2 T - 24 T^{2} - 100 T^{3} + 439 T^{4} + 2546 T^{5} + 6524 T^{6} - 38396 T^{7} - 127700 T^{8} - 38396 p T^{9} + 6524 p^{2} T^{10} + 2546 p^{3} T^{11} + 439 p^{4} T^{12} - 100 p^{5} T^{13} - 24 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 19 | \( 1 - 36 T^{2} - 244 T^{3} + 739 T^{4} + 6710 T^{5} + 1096 p T^{6} - 98942 T^{7} - 570724 T^{8} - 98942 p T^{9} + 1096 p^{3} T^{10} + 6710 p^{3} T^{11} + 739 p^{4} T^{12} - 244 p^{5} T^{13} - 36 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( 1 - 4 T - 34 T^{2} + 200 T^{3} - 7 T^{4} - 944 T^{5} - 10850 T^{6} - 35868 T^{7} + 780852 T^{8} - 35868 p T^{9} - 10850 p^{2} T^{10} - 944 p^{3} T^{11} - 7 p^{4} T^{12} + 200 p^{5} T^{13} - 34 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 8 T + 128 T^{2} + 682 T^{3} + 5745 T^{4} + 682 p T^{5} + 128 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 - 12 T + 28 T^{2} - 160 T^{3} + 2190 T^{4} + 2424 T^{5} - 76128 T^{6} + 348684 T^{7} - 2030189 T^{8} + 348684 p T^{9} - 76128 p^{2} T^{10} + 2424 p^{3} T^{11} + 2190 p^{4} T^{12} - 160 p^{5} T^{13} + 28 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 37 | \( 1 - 4 T - 122 T^{2} + 336 T^{3} + 9793 T^{4} - 17900 T^{5} - 522282 T^{6} + 6944 p T^{7} + 22257092 T^{8} + 6944 p^{2} T^{9} - 522282 p^{2} T^{10} - 17900 p^{3} T^{11} + 9793 p^{4} T^{12} + 336 p^{5} T^{13} - 122 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 41 | \( ( 1 + 2 T + 120 T^{2} + 310 T^{3} + 6538 T^{4} + 310 p T^{5} + 120 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 18 T + 232 T^{2} - 2014 T^{3} + 14869 T^{4} - 2014 p T^{5} + 232 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 12 T - 82 T^{2} + 16 p T^{3} + 13457 T^{4} - 68796 T^{5} - 783266 T^{6} + 399704 T^{7} + 54619956 T^{8} + 399704 p T^{9} - 783266 p^{2} T^{10} - 68796 p^{3} T^{11} + 13457 p^{4} T^{12} + 16 p^{6} T^{13} - 82 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 53 | \( 1 + 12 T + 56 T^{2} - 832 T^{3} - 13546 T^{4} - 105984 T^{5} - 91520 T^{6} + 5351324 T^{7} + 65244435 T^{8} + 5351324 p T^{9} - 91520 p^{2} T^{10} - 105984 p^{3} T^{11} - 13546 p^{4} T^{12} - 832 p^{5} T^{13} + 56 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 59 | \( 1 - 12 T - 74 T^{2} + 1208 T^{3} + 5569 T^{4} - 73360 T^{5} - 324634 T^{6} + 1264412 T^{7} + 27329972 T^{8} + 1264412 p T^{9} - 324634 p^{2} T^{10} - 73360 p^{3} T^{11} + 5569 p^{4} T^{12} + 1208 p^{5} T^{13} - 74 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( 1 + 2 T - 216 T^{2} - 228 T^{3} + 28314 T^{4} + 16182 T^{5} - 2563912 T^{6} - 387746 T^{7} + 178127415 T^{8} - 387746 p T^{9} - 2563912 p^{2} T^{10} + 16182 p^{3} T^{11} + 28314 p^{4} T^{12} - 228 p^{5} T^{13} - 216 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 67 | \( 1 + 28 T + 324 T^{2} + 1888 T^{3} + 6510 T^{4} + 29080 T^{5} + 472480 T^{6} + 8872436 T^{7} + 99414595 T^{8} + 8872436 p T^{9} + 472480 p^{2} T^{10} + 29080 p^{3} T^{11} + 6510 p^{4} T^{12} + 1888 p^{5} T^{13} + 324 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 12 T + 286 T^{2} + 2260 T^{3} + 29831 T^{4} + 2260 p T^{5} + 286 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 + 6 T + 16 T^{2} - 1084 T^{3} - 12630 T^{4} - 57030 T^{5} + 123528 T^{6} + 6885378 T^{7} + 59722615 T^{8} + 6885378 p T^{9} + 123528 p^{2} T^{10} - 57030 p^{3} T^{11} - 12630 p^{4} T^{12} - 1084 p^{5} T^{13} + 16 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 79 | \( 1 + 2 T - 164 T^{2} - 1476 T^{3} + 11962 T^{4} + 165982 T^{5} + 347064 T^{6} - 7874110 T^{7} - 67195297 T^{8} - 7874110 p T^{9} + 347064 p^{2} T^{10} + 165982 p^{3} T^{11} + 11962 p^{4} T^{12} - 1476 p^{5} T^{13} - 164 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 83 | \( ( 1 - 12 T + 236 T^{2} - 2044 T^{3} + 27990 T^{4} - 2044 p T^{5} + 236 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 - 8 T - 60 T^{2} + 848 T^{3} - 10262 T^{4} + 96488 T^{5} - 110704 T^{6} - 10006904 T^{7} + 164456179 T^{8} - 10006904 p T^{9} - 110704 p^{2} T^{10} + 96488 p^{3} T^{11} - 10262 p^{4} T^{12} + 848 p^{5} T^{13} - 60 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 97 | \( ( 1 + 44 T + 1070 T^{2} + 17084 T^{3} + 197263 T^{4} + 17084 p T^{5} + 1070 p^{2} T^{6} + 44 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.60639343145168684996990703334, −4.50770376737198839066963067656, −4.43353051433278972174826423443, −4.13188685754835853126872866666, −4.08827073955024393522000824196, −3.78819041028867657200344784851, −3.70788911283629576013094729562, −3.50979432976736080367461554765, −3.46003492026133032423989807151, −3.15325825105246043716377920074, −2.97729711060326083206686738474, −2.84509937878824010384602672624, −2.83902839336353588532974381923, −2.77314863342186153824634007632, −2.54335474407397683004359898698, −2.28225935909294695506267709265, −2.19891043181301778298131367874, −2.18272001752226645472511829522, −2.14157014486405603342395612558, −1.67594575944611683732055197894, −1.41101548088381238794483599754, −1.14846418186909967931224067904, −0.971547334627725741580565518022, −0.920240972329514785461180459295, −0.13425288662987233237221666298,
0.13425288662987233237221666298, 0.920240972329514785461180459295, 0.971547334627725741580565518022, 1.14846418186909967931224067904, 1.41101548088381238794483599754, 1.67594575944611683732055197894, 2.14157014486405603342395612558, 2.18272001752226645472511829522, 2.19891043181301778298131367874, 2.28225935909294695506267709265, 2.54335474407397683004359898698, 2.77314863342186153824634007632, 2.83902839336353588532974381923, 2.84509937878824010384602672624, 2.97729711060326083206686738474, 3.15325825105246043716377920074, 3.46003492026133032423989807151, 3.50979432976736080367461554765, 3.70788911283629576013094729562, 3.78819041028867657200344784851, 4.08827073955024393522000824196, 4.13188685754835853126872866666, 4.43353051433278972174826423443, 4.50770376737198839066963067656, 4.60639343145168684996990703334
Plot not available for L-functions of degree greater than 10.
