L(s) = 1 | + 12·2-s + 78·4-s + 5·5-s + 6·7-s + 364·8-s + 60·10-s + 6·11-s + 3·13-s + 72·14-s + 1.36e3·16-s + 6·17-s + 8·19-s + 390·20-s + 72·22-s + 11·23-s − 12·25-s + 36·26-s + 468·28-s + 29·29-s + 2·31-s + 4.36e3·32-s + 72·34-s + 30·35-s + 5·37-s + 96·38-s + 1.82e3·40-s + 22·41-s + ⋯ |
L(s) = 1 | + 8.48·2-s + 39·4-s + 2.23·5-s + 2.26·7-s + 128.·8-s + 18.9·10-s + 1.80·11-s + 0.832·13-s + 19.2·14-s + 341.·16-s + 1.45·17-s + 1.83·19-s + 87.2·20-s + 15.3·22-s + 2.29·23-s − 2.39·25-s + 7.06·26-s + 88.4·28-s + 5.38·29-s + 0.359·31-s + 772.·32-s + 12.3·34-s + 5.07·35-s + 0.821·37-s + 15.5·38-s + 287.·40-s + 3.43·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{36} \cdot 149^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{36} \cdot 149^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.301624304\times10^{6}\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.301624304\times10^{6}\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( ( 1 - T )^{12} \) |
| 3 | \( 1 \) |
| 149 | \( ( 1 + T )^{12} \) |
good | 5 | \( 1 - p T + 37 T^{2} - 133 T^{3} + 604 T^{4} - 1787 T^{5} + 6392 T^{6} - 16572 T^{7} + 50924 T^{8} - 119058 T^{9} + 65492 p T^{10} - 701969 T^{11} + 1770847 T^{12} - 701969 p T^{13} + 65492 p^{3} T^{14} - 119058 p^{3} T^{15} + 50924 p^{4} T^{16} - 16572 p^{5} T^{17} + 6392 p^{6} T^{18} - 1787 p^{7} T^{19} + 604 p^{8} T^{20} - 133 p^{9} T^{21} + 37 p^{10} T^{22} - p^{12} T^{23} + p^{12} T^{24} \) |
| 7 | \( 1 - 6 T + 53 T^{2} - 242 T^{3} + 1336 T^{4} - 5118 T^{5} + 22241 T^{6} - 74476 T^{7} + 275221 T^{8} - 117375 p T^{9} + 2668448 T^{10} - 7159148 T^{11} + 20764979 T^{12} - 7159148 p T^{13} + 2668448 p^{2} T^{14} - 117375 p^{4} T^{15} + 275221 p^{4} T^{16} - 74476 p^{5} T^{17} + 22241 p^{6} T^{18} - 5118 p^{7} T^{19} + 1336 p^{8} T^{20} - 242 p^{9} T^{21} + 53 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) |
| 11 | \( 1 - 6 T + 83 T^{2} - 400 T^{3} + 3275 T^{4} - 13795 T^{5} + 85490 T^{6} - 324432 T^{7} + 1657369 T^{8} - 5718173 T^{9} + 25187716 T^{10} - 78968378 T^{11} + 307925583 T^{12} - 78968378 p T^{13} + 25187716 p^{2} T^{14} - 5718173 p^{3} T^{15} + 1657369 p^{4} T^{16} - 324432 p^{5} T^{17} + 85490 p^{6} T^{18} - 13795 p^{7} T^{19} + 3275 p^{8} T^{20} - 400 p^{9} T^{21} + 83 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) |
| 13 | \( 1 - 3 T + 93 T^{2} - 185 T^{3} + 4117 T^{4} - 5292 T^{5} + 120357 T^{6} - 96192 T^{7} + 2648158 T^{8} - 100457 p T^{9} + 46450531 T^{10} - 15709542 T^{11} + 665668655 T^{12} - 15709542 p T^{13} + 46450531 p^{2} T^{14} - 100457 p^{4} T^{15} + 2648158 p^{4} T^{16} - 96192 p^{5} T^{17} + 120357 p^{6} T^{18} - 5292 p^{7} T^{19} + 4117 p^{8} T^{20} - 185 p^{9} T^{21} + 93 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) |
| 17 | \( 1 - 6 T + 105 T^{2} - 597 T^{3} + 6053 T^{4} - 32404 T^{5} + 242061 T^{6} - 1194171 T^{7} + 7302766 T^{8} - 32901614 T^{9} + 173184434 T^{10} - 705647403 T^{11} + 3282052285 T^{12} - 705647403 p T^{13} + 173184434 p^{2} T^{14} - 32901614 p^{3} T^{15} + 7302766 p^{4} T^{16} - 1194171 p^{5} T^{17} + 242061 p^{6} T^{18} - 32404 p^{7} T^{19} + 6053 p^{8} T^{20} - 597 p^{9} T^{21} + 105 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) |
| 19 | \( 1 - 8 T + 129 T^{2} - 782 T^{3} + 7183 T^{4} - 38111 T^{5} + 260210 T^{6} - 1339878 T^{7} + 7370283 T^{8} - 37496113 T^{9} + 172035206 T^{10} - 849459602 T^{11} + 3458078643 T^{12} - 849459602 p T^{13} + 172035206 p^{2} T^{14} - 37496113 p^{3} T^{15} + 7370283 p^{4} T^{16} - 1339878 p^{5} T^{17} + 260210 p^{6} T^{18} - 38111 p^{7} T^{19} + 7183 p^{8} T^{20} - 782 p^{9} T^{21} + 129 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) |
| 23 | \( 1 - 11 T + 262 T^{2} - 2269 T^{3} + 30685 T^{4} - 221080 T^{5} + 2185501 T^{6} - 13469165 T^{7} + 106742913 T^{8} - 570337343 T^{9} + 164684779 p T^{10} - 17621235363 T^{11} + 100387404119 T^{12} - 17621235363 p T^{13} + 164684779 p^{3} T^{14} - 570337343 p^{3} T^{15} + 106742913 p^{4} T^{16} - 13469165 p^{5} T^{17} + 2185501 p^{6} T^{18} - 221080 p^{7} T^{19} + 30685 p^{8} T^{20} - 2269 p^{9} T^{21} + 262 p^{10} T^{22} - 11 p^{11} T^{23} + p^{12} T^{24} \) |
| 29 | \( 1 - p T + 21 p T^{2} - 9253 T^{3} + 117590 T^{4} - 1265253 T^{5} + 12043636 T^{6} - 102332654 T^{7} + 791357894 T^{8} - 5594431404 T^{9} + 36508906376 T^{10} - 220065448467 T^{11} + 1231373716743 T^{12} - 220065448467 p T^{13} + 36508906376 p^{2} T^{14} - 5594431404 p^{3} T^{15} + 791357894 p^{4} T^{16} - 102332654 p^{5} T^{17} + 12043636 p^{6} T^{18} - 1265253 p^{7} T^{19} + 117590 p^{8} T^{20} - 9253 p^{9} T^{21} + 21 p^{11} T^{22} - p^{12} T^{23} + p^{12} T^{24} \) |
| 31 | \( 1 - 2 T + 316 T^{2} - 629 T^{3} + 47111 T^{4} - 90807 T^{5} + 4384473 T^{6} - 7978244 T^{7} + 283776859 T^{8} - 475465865 T^{9} + 13457602963 T^{10} - 20221270731 T^{11} + 479490551393 T^{12} - 20221270731 p T^{13} + 13457602963 p^{2} T^{14} - 475465865 p^{3} T^{15} + 283776859 p^{4} T^{16} - 7978244 p^{5} T^{17} + 4384473 p^{6} T^{18} - 90807 p^{7} T^{19} + 47111 p^{8} T^{20} - 629 p^{9} T^{21} + 316 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) |
| 37 | \( 1 - 5 T + 119 T^{2} - 115 T^{3} + 7645 T^{4} - 1649 T^{5} + 440841 T^{6} + 321802 T^{7} + 19993330 T^{8} + 6656905 T^{9} + 950071077 T^{10} + 125428093 T^{11} + 34857558493 T^{12} + 125428093 p T^{13} + 950071077 p^{2} T^{14} + 6656905 p^{3} T^{15} + 19993330 p^{4} T^{16} + 321802 p^{5} T^{17} + 440841 p^{6} T^{18} - 1649 p^{7} T^{19} + 7645 p^{8} T^{20} - 115 p^{9} T^{21} + 119 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \) |
| 41 | \( 1 - 22 T + 570 T^{2} - 8654 T^{3} + 134326 T^{4} - 1580751 T^{5} + 18404440 T^{6} - 177542004 T^{7} + 1677920778 T^{8} - 13669761819 T^{9} + 108769892738 T^{10} - 759716186158 T^{11} + 5178296674430 T^{12} - 759716186158 p T^{13} + 108769892738 p^{2} T^{14} - 13669761819 p^{3} T^{15} + 1677920778 p^{4} T^{16} - 177542004 p^{5} T^{17} + 18404440 p^{6} T^{18} - 1580751 p^{7} T^{19} + 134326 p^{8} T^{20} - 8654 p^{9} T^{21} + 570 p^{10} T^{22} - 22 p^{11} T^{23} + p^{12} T^{24} \) |
| 43 | \( 1 - 9 T + 340 T^{2} - 3067 T^{3} + 59853 T^{4} - 506558 T^{5} + 6943178 T^{6} - 53812233 T^{7} + 581760563 T^{8} - 4070801777 T^{9} + 36810465985 T^{10} - 229950733232 T^{11} + 1796356034379 T^{12} - 229950733232 p T^{13} + 36810465985 p^{2} T^{14} - 4070801777 p^{3} T^{15} + 581760563 p^{4} T^{16} - 53812233 p^{5} T^{17} + 6943178 p^{6} T^{18} - 506558 p^{7} T^{19} + 59853 p^{8} T^{20} - 3067 p^{9} T^{21} + 340 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) |
| 47 | \( 1 - 15 T + 362 T^{2} - 4070 T^{3} + 57742 T^{4} - 536591 T^{5} + 5843321 T^{6} - 47584927 T^{7} + 441055404 T^{8} - 3260908938 T^{9} + 26980598283 T^{10} - 183650618078 T^{11} + 1382094670383 T^{12} - 183650618078 p T^{13} + 26980598283 p^{2} T^{14} - 3260908938 p^{3} T^{15} + 441055404 p^{4} T^{16} - 47584927 p^{5} T^{17} + 5843321 p^{6} T^{18} - 536591 p^{7} T^{19} + 57742 p^{8} T^{20} - 4070 p^{9} T^{21} + 362 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) |
| 53 | \( 1 - 12 T + 401 T^{2} - 4001 T^{3} + 1331 p T^{4} - 575383 T^{5} + 7018763 T^{6} - 45343642 T^{7} + 427988438 T^{8} - 2060058947 T^{9} + 17268440727 T^{10} - 61388213430 T^{11} + 693724505451 T^{12} - 61388213430 p T^{13} + 17268440727 p^{2} T^{14} - 2060058947 p^{3} T^{15} + 427988438 p^{4} T^{16} - 45343642 p^{5} T^{17} + 7018763 p^{6} T^{18} - 575383 p^{7} T^{19} + 1331 p^{9} T^{20} - 4001 p^{9} T^{21} + 401 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) |
| 59 | \( 1 - 34 T + 991 T^{2} - 19524 T^{3} + 5778 p T^{4} - 4900505 T^{5} + 64248442 T^{6} - 743642222 T^{7} + 7996728095 T^{8} - 78396120401 T^{9} + 720435496810 T^{10} - 6116626005439 T^{11} + 48799744500633 T^{12} - 6116626005439 p T^{13} + 720435496810 p^{2} T^{14} - 78396120401 p^{3} T^{15} + 7996728095 p^{4} T^{16} - 743642222 p^{5} T^{17} + 64248442 p^{6} T^{18} - 4900505 p^{7} T^{19} + 5778 p^{9} T^{20} - 19524 p^{9} T^{21} + 991 p^{10} T^{22} - 34 p^{11} T^{23} + p^{12} T^{24} \) |
| 61 | \( 1 + 4 T + 406 T^{2} + 2298 T^{3} + 82658 T^{4} + 590587 T^{5} + 185243 p T^{6} + 92930194 T^{7} + 1165100774 T^{8} + 10151363202 T^{9} + 95808777573 T^{10} + 817946848303 T^{11} + 6441572717915 T^{12} + 817946848303 p T^{13} + 95808777573 p^{2} T^{14} + 10151363202 p^{3} T^{15} + 1165100774 p^{4} T^{16} + 92930194 p^{5} T^{17} + 185243 p^{7} T^{18} + 590587 p^{7} T^{19} + 82658 p^{8} T^{20} + 2298 p^{9} T^{21} + 406 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) |
| 67 | \( 1 - T + 257 T^{2} + 45 T^{3} + 40286 T^{4} + 24251 T^{5} + 5084197 T^{6} + 4140364 T^{7} + 503637946 T^{8} + 544064397 T^{9} + 42191060791 T^{10} + 45687110740 T^{11} + 3079301020221 T^{12} + 45687110740 p T^{13} + 42191060791 p^{2} T^{14} + 544064397 p^{3} T^{15} + 503637946 p^{4} T^{16} + 4140364 p^{5} T^{17} + 5084197 p^{6} T^{18} + 24251 p^{7} T^{19} + 40286 p^{8} T^{20} + 45 p^{9} T^{21} + 257 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) |
| 71 | \( 1 - 21 T + 815 T^{2} - 13439 T^{3} + 298431 T^{4} - 4057110 T^{5} + 66280222 T^{6} - 764199396 T^{7} + 10056987962 T^{8} - 99917777007 T^{9} + 1105322506763 T^{10} - 9528643307964 T^{11} + 90523489032239 T^{12} - 9528643307964 p T^{13} + 1105322506763 p^{2} T^{14} - 99917777007 p^{3} T^{15} + 10056987962 p^{4} T^{16} - 764199396 p^{5} T^{17} + 66280222 p^{6} T^{18} - 4057110 p^{7} T^{19} + 298431 p^{8} T^{20} - 13439 p^{9} T^{21} + 815 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \) |
| 73 | \( 1 + 2 T + 398 T^{2} + 1673 T^{3} + 77385 T^{4} + 460592 T^{5} + 10426140 T^{6} + 69484139 T^{7} + 1135186987 T^{8} + 7189648889 T^{9} + 104932563821 T^{10} + 595825738667 T^{11} + 8285671901785 T^{12} + 595825738667 p T^{13} + 104932563821 p^{2} T^{14} + 7189648889 p^{3} T^{15} + 1135186987 p^{4} T^{16} + 69484139 p^{5} T^{17} + 10426140 p^{6} T^{18} + 460592 p^{7} T^{19} + 77385 p^{8} T^{20} + 1673 p^{9} T^{21} + 398 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) |
| 79 | \( 1 - 9 T + 518 T^{2} - 3691 T^{3} + 131738 T^{4} - 781364 T^{5} + 22668483 T^{6} - 116720572 T^{7} + 2981526684 T^{8} - 13678163642 T^{9} + 315293662403 T^{10} - 1306179035946 T^{11} + 27391779560869 T^{12} - 1306179035946 p T^{13} + 315293662403 p^{2} T^{14} - 13678163642 p^{3} T^{15} + 2981526684 p^{4} T^{16} - 116720572 p^{5} T^{17} + 22668483 p^{6} T^{18} - 781364 p^{7} T^{19} + 131738 p^{8} T^{20} - 3691 p^{9} T^{21} + 518 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) |
| 83 | \( 1 - 10 T + 595 T^{2} - 4428 T^{3} + 153376 T^{4} - 876647 T^{5} + 24166252 T^{6} - 117887750 T^{7} + 34644621 p T^{8} - 14127122419 T^{9} + 295311784318 T^{10} - 1507783663471 T^{11} + 26514609855899 T^{12} - 1507783663471 p T^{13} + 295311784318 p^{2} T^{14} - 14127122419 p^{3} T^{15} + 34644621 p^{5} T^{16} - 117887750 p^{5} T^{17} + 24166252 p^{6} T^{18} - 876647 p^{7} T^{19} + 153376 p^{8} T^{20} - 4428 p^{9} T^{21} + 595 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) |
| 89 | \( 1 + 2 T + 662 T^{2} + 1003 T^{3} + 214775 T^{4} + 220301 T^{5} + 45647850 T^{6} + 27386804 T^{7} + 7148082988 T^{8} + 2124959770 T^{9} + 875420427455 T^{10} + 125970749265 T^{11} + 86394150305655 T^{12} + 125970749265 p T^{13} + 875420427455 p^{2} T^{14} + 2124959770 p^{3} T^{15} + 7148082988 p^{4} T^{16} + 27386804 p^{5} T^{17} + 45647850 p^{6} T^{18} + 220301 p^{7} T^{19} + 214775 p^{8} T^{20} + 1003 p^{9} T^{21} + 662 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) |
| 97 | \( 1 + 13 T + 909 T^{2} + 10619 T^{3} + 399660 T^{4} + 4192016 T^{5} + 111664562 T^{6} + 1049739005 T^{7} + 22001910224 T^{8} + 184572841939 T^{9} + 3213928058406 T^{10} + 23874378565506 T^{11} + 356479471151649 T^{12} + 23874378565506 p T^{13} + 3213928058406 p^{2} T^{14} + 184572841939 p^{3} T^{15} + 22001910224 p^{4} T^{16} + 1049739005 p^{5} T^{17} + 111664562 p^{6} T^{18} + 4192016 p^{7} T^{19} + 399660 p^{8} T^{20} + 10619 p^{9} T^{21} + 909 p^{10} T^{22} + 13 p^{11} T^{23} + p^{12} T^{24} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.44939940992169008271315659063, −2.31847701790704439968011748912, −2.09945992316997258607393051331, −2.09482524120551535034541666721, −2.04117907986972047111573735830, −1.94469024283679965532531265221, −1.89200885746742275281247954051, −1.87813634381965607627039207557, −1.83985419762535011513018666156, −1.81858297641113557903018028533, −1.73266670773984473526015579423, −1.66922891221371481565432826486, −1.66402533008116796915969199998, −1.19050477223277177421934249060, −1.18239525789331925847751010473, −1.17638443086114042048317806996, −1.07788352999547408450728575655, −0.993955458207582497334215173889, −0.985815341242616477174204183899, −0.930029824180469915767319064702, −0.810617520397269001290670103114, −0.797231920098640731379485398540, −0.70315539051059376390896930573, −0.55768597622482845535817162715, −0.26877334130636803071846390598,
0.26877334130636803071846390598, 0.55768597622482845535817162715, 0.70315539051059376390896930573, 0.797231920098640731379485398540, 0.810617520397269001290670103114, 0.930029824180469915767319064702, 0.985815341242616477174204183899, 0.993955458207582497334215173889, 1.07788352999547408450728575655, 1.17638443086114042048317806996, 1.18239525789331925847751010473, 1.19050477223277177421934249060, 1.66402533008116796915969199998, 1.66922891221371481565432826486, 1.73266670773984473526015579423, 1.81858297641113557903018028533, 1.83985419762535011513018666156, 1.87813634381965607627039207557, 1.89200885746742275281247954051, 1.94469024283679965532531265221, 2.04117907986972047111573735830, 2.09482524120551535034541666721, 2.09945992316997258607393051331, 2.31847701790704439968011748912, 2.44939940992169008271315659063
Plot not available for L-functions of degree greater than 10.