L(s) = 1 | − 2-s + 3·3-s + 5·4-s − 3·5-s − 3·6-s + 3·7-s − 2·8-s + 10·9-s + 3·10-s + 2·11-s + 15·12-s + 3·13-s − 3·14-s − 9·15-s + 14·16-s + 2·17-s − 10·18-s − 5·19-s − 15·20-s + 9·21-s − 2·22-s − 24·23-s − 6·24-s + 8·25-s − 3·26-s + 24·27-s + 15·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.73·3-s + 5/2·4-s − 1.34·5-s − 1.22·6-s + 1.13·7-s − 0.707·8-s + 10/3·9-s + 0.948·10-s + 0.603·11-s + 4.33·12-s + 0.832·13-s − 0.801·14-s − 2.32·15-s + 7/2·16-s + 0.485·17-s − 2.35·18-s − 1.14·19-s − 3.35·20-s + 1.96·21-s − 0.426·22-s − 5.00·23-s − 1.22·24-s + 8/5·25-s − 0.588·26-s + 4.61·27-s + 2.83·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.977835050\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.977835050\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( ( 1 - T - 9 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
good | 2 | \( 1 + T - p^{2} T^{2} - 7 T^{3} + T^{4} + 23 T^{5} + 15 p T^{6} - 25 T^{7} - 93 T^{8} - 25 p T^{9} + 15 p^{3} T^{10} + 23 p^{3} T^{11} + p^{4} T^{12} - 7 p^{5} T^{13} - p^{8} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) |
| 3 | \( 1 - p T - T^{2} + p^{2} T^{3} + T^{4} - 8 p T^{5} + 20 T^{6} + 20 p T^{7} - 203 T^{8} + 20 p^{2} T^{9} + 20 p^{2} T^{10} - 8 p^{4} T^{11} + p^{4} T^{12} + p^{7} T^{13} - p^{6} T^{14} - p^{8} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 + 3 T + T^{2} - T^{3} + 39 T^{4} + 22 T^{5} - 296 T^{6} - 434 T^{7} + 191 T^{8} - 434 p T^{9} - 296 p^{2} T^{10} + 22 p^{3} T^{11} + 39 p^{4} T^{12} - p^{5} T^{13} + p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 - 3 T - 3 T^{2} + 24 T^{3} + 3 T^{4} - 24 T^{5} - 206 T^{6} - 573 T^{7} + 5433 T^{8} - 573 p T^{9} - 206 p^{2} T^{10} - 24 p^{3} T^{11} + 3 p^{4} T^{12} + 24 p^{5} T^{13} - 3 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 13 | \( 1 - 3 T - 21 T^{2} + 79 T^{3} - 9 T^{4} - 834 T^{5} + 4250 T^{6} + 3830 T^{7} - 70373 T^{8} + 3830 p T^{9} + 4250 p^{2} T^{10} - 834 p^{3} T^{11} - 9 p^{4} T^{12} + 79 p^{5} T^{13} - 21 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 19 | \( 1 + 5 T + 39 T^{2} + 235 T^{3} + 1485 T^{4} + 7480 T^{5} + 39966 T^{6} + 170930 T^{7} + 865649 T^{8} + 170930 p T^{9} + 39966 p^{2} T^{10} + 7480 p^{3} T^{11} + 1485 p^{4} T^{12} + 235 p^{5} T^{13} + 39 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 23 | \( ( 1 + 12 T + 4 p T^{2} + 572 T^{3} + 2918 T^{4} + 572 p T^{5} + 4 p^{3} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( 1 - 10 T + 50 T^{2} - 440 T^{3} + 4099 T^{4} - 23450 T^{5} + 122220 T^{6} - 844400 T^{7} + 5206461 T^{8} - 844400 p T^{9} + 122220 p^{2} T^{10} - 23450 p^{3} T^{11} + 4099 p^{4} T^{12} - 440 p^{5} T^{13} + 50 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) |
| 31 | \( 1 - 17 T + 58 T^{2} + 794 T^{3} - 6838 T^{4} - 5975 T^{5} + 222692 T^{6} - 70072 T^{7} - 5919725 T^{8} - 70072 p T^{9} + 222692 p^{2} T^{10} - 5975 p^{3} T^{11} - 6838 p^{4} T^{12} + 794 p^{5} T^{13} + 58 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} \) |
| 37 | \( 1 + 13 T + 56 T^{2} + 65 T^{3} + 306 T^{4} - 656 T^{5} - 56700 T^{6} - 708706 T^{7} - 5210633 T^{8} - 708706 p T^{9} - 56700 p^{2} T^{10} - 656 p^{3} T^{11} + 306 p^{4} T^{12} + 65 p^{5} T^{13} + 56 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \) |
| 41 | \( 1 + 22 T + 4 p T^{2} + 590 T^{3} + 4853 T^{4} + 47270 T^{5} + 233976 T^{6} + 1558588 T^{7} + 13143513 T^{8} + 1558588 p T^{9} + 233976 p^{2} T^{10} + 47270 p^{3} T^{11} + 4853 p^{4} T^{12} + 590 p^{5} T^{13} + 4 p^{7} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) |
| 43 | \( ( 1 - 4 T + 144 T^{2} - 377 T^{3} + 8577 T^{4} - 377 p T^{5} + 144 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 9 T + 70 T^{2} + 106 T^{3} - 4506 T^{4} + 38633 T^{5} - 81194 T^{6} - 1740028 T^{7} + 15746231 T^{8} - 1740028 p T^{9} - 81194 p^{2} T^{10} + 38633 p^{3} T^{11} - 4506 p^{4} T^{12} + 106 p^{5} T^{13} + 70 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) |
| 53 | \( 1 + 23 T + 307 T^{2} + 2874 T^{3} + 20701 T^{4} + 87608 T^{5} - 88520 T^{6} - 5617183 T^{7} - 53794401 T^{8} - 5617183 p T^{9} - 88520 p^{2} T^{10} + 87608 p^{3} T^{11} + 20701 p^{4} T^{12} + 2874 p^{5} T^{13} + 307 p^{6} T^{14} + 23 p^{7} T^{15} + p^{8} T^{16} \) |
| 59 | \( 1 + 35 T + 604 T^{2} + 7245 T^{3} + 74610 T^{4} + 744230 T^{5} + 6991266 T^{6} + 59177710 T^{7} + 464835829 T^{8} + 59177710 p T^{9} + 6991266 p^{2} T^{10} + 744230 p^{3} T^{11} + 74610 p^{4} T^{12} + 7245 p^{5} T^{13} + 604 p^{6} T^{14} + 35 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( 1 + 19 T + 48 T^{2} - 1201 T^{3} - 7968 T^{4} + 6470 T^{5} + 72522 T^{6} - 130658 T^{7} + 2804795 T^{8} - 130658 p T^{9} + 72522 p^{2} T^{10} + 6470 p^{3} T^{11} - 7968 p^{4} T^{12} - 1201 p^{5} T^{13} + 48 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \) |
| 67 | \( ( 1 + 5 T + 258 T^{2} + 955 T^{3} + 25569 T^{4} + 955 p T^{5} + 258 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( 1 + 5 T + 90 T^{2} + 930 T^{3} + 10574 T^{4} + 56475 T^{5} + 744050 T^{6} + 5166800 T^{7} + 38102511 T^{8} + 5166800 p T^{9} + 744050 p^{2} T^{10} + 56475 p^{3} T^{11} + 10574 p^{4} T^{12} + 930 p^{5} T^{13} + 90 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 73 | \( 1 - 39 T + 573 T^{2} - 3887 T^{3} + 13623 T^{4} - 63102 T^{5} + 1427096 T^{6} - 33926386 T^{7} + 415500883 T^{8} - 33926386 p T^{9} + 1427096 p^{2} T^{10} - 63102 p^{3} T^{11} + 13623 p^{4} T^{12} - 3887 p^{5} T^{13} + 573 p^{6} T^{14} - 39 p^{7} T^{15} + p^{8} T^{16} \) |
| 79 | \( 1 + 3 T + 55 T^{2} + 1140 T^{3} + 55 p T^{4} + 150784 T^{5} + 1157722 T^{6} + 6681605 T^{7} + 114816665 T^{8} + 6681605 p T^{9} + 1157722 p^{2} T^{10} + 150784 p^{3} T^{11} + 55 p^{5} T^{12} + 1140 p^{5} T^{13} + 55 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 83 | \( 1 - 29 T + 309 T^{2} - 815 T^{3} - 11149 T^{4} + 38438 T^{5} + 1549080 T^{6} - 22799578 T^{7} + 210737177 T^{8} - 22799578 p T^{9} + 1549080 p^{2} T^{10} + 38438 p^{3} T^{11} - 11149 p^{4} T^{12} - 815 p^{5} T^{13} + 309 p^{6} T^{14} - 29 p^{7} T^{15} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 20 T + 390 T^{2} - 4155 T^{3} + 48407 T^{4} - 4155 p T^{5} + 390 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 + 5 T - 89 T^{2} + 1030 T^{3} + 9287 T^{4} - 113930 T^{5} + 1213208 T^{6} + 13189775 T^{7} - 95415245 T^{8} + 13189775 p T^{9} + 1213208 p^{2} T^{10} - 113930 p^{3} T^{11} + 9287 p^{4} T^{12} + 1030 p^{5} T^{13} - 89 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) |
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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
−5.99667222373700420056163202343, −5.99235699952214755470630018908, −5.30051348619421521043856270681, −5.09411631345531722000967641010, −4.96828775305094729165418315310, −4.94859926246064495468579477534, −4.88482437381033306722755597890, −4.45399273309801652446671189282, −4.40241285913152671277975203443, −4.27066975831041600302913018118, −4.09973668058138523619272629035, −3.82300156514752023183145739167, −3.73070686166434267150191672902, −3.37504454938035191380108371659, −3.32336578078363823701008892701, −3.25213666082050406454401537822, −3.19771637676085693303937277010, −2.50233172518530709789166891019, −2.46097480371085671956976166277, −2.19263888269656206054499470956, −1.94168929167169809105559901415, −1.71446039578737199900129114519, −1.71306109526390893919468878460, −1.31123718880300132559909384341, −1.09635310128844823176953325476,
1.09635310128844823176953325476, 1.31123718880300132559909384341, 1.71306109526390893919468878460, 1.71446039578737199900129114519, 1.94168929167169809105559901415, 2.19263888269656206054499470956, 2.46097480371085671956976166277, 2.50233172518530709789166891019, 3.19771637676085693303937277010, 3.25213666082050406454401537822, 3.32336578078363823701008892701, 3.37504454938035191380108371659, 3.73070686166434267150191672902, 3.82300156514752023183145739167, 4.09973668058138523619272629035, 4.27066975831041600302913018118, 4.40241285913152671277975203443, 4.45399273309801652446671189282, 4.88482437381033306722755597890, 4.94859926246064495468579477534, 4.96828775305094729165418315310, 5.09411631345531722000967641010, 5.30051348619421521043856270681, 5.99235699952214755470630018908, 5.99667222373700420056163202343
Plot not available for L-functions of degree greater than 10.