L(s) = 1 | − 2-s + 2·4-s + 5·5-s + 4·7-s + 8-s − 5·10-s − 10·11-s + 8·13-s − 4·14-s + 16-s + 10·17-s + 5·19-s + 10·20-s + 10·22-s − 2·23-s + 10·25-s − 8·26-s + 8·28-s − 7·29-s − 18·31-s − 10·34-s + 20·35-s + 12·37-s − 5·38-s + 5·40-s + 12·41-s + 8·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 4-s + 2.23·5-s + 1.51·7-s + 0.353·8-s − 1.58·10-s − 3.01·11-s + 2.21·13-s − 1.06·14-s + 1/4·16-s + 2.42·17-s + 1.14·19-s + 2.23·20-s + 2.13·22-s − 0.417·23-s + 2·25-s − 1.56·26-s + 1.51·28-s − 1.29·29-s − 3.23·31-s − 1.71·34-s + 3.38·35-s + 1.97·37-s − 0.811·38-s + 0.790·40-s + 1.87·41-s + 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 5^{10} \cdot 19^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 5^{10} \cdot 19^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.980830340\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.980830340\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( ( 1 - T + T^{2} )^{5} \) |
| 19 | \( 1 - 5 T + 18 T^{2} + 21 T^{3} + 501 T^{4} - 2376 T^{5} + 501 p T^{6} + 21 p^{2} T^{7} + 18 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \) |
good | 2 | \( 1 + T - T^{2} - p^{2} T^{3} - p^{2} T^{4} + p^{2} T^{5} + 11 T^{6} + 5 T^{7} - 11 T^{8} - 3 p^{2} T^{9} - 3 p T^{10} - 3 p^{3} T^{11} - 11 p^{2} T^{12} + 5 p^{3} T^{13} + 11 p^{4} T^{14} + p^{7} T^{15} - p^{8} T^{16} - p^{9} T^{17} - p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) |
| 7 | \( ( 1 - 2 T + 12 T^{2} - 20 T^{3} + 79 T^{4} - 60 T^{5} + 79 p T^{6} - 20 p^{2} T^{7} + 12 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 11 | \( ( 1 + 5 T + 23 T^{2} + 72 T^{3} + 410 T^{4} + 1366 T^{5} + 410 p T^{6} + 72 p^{2} T^{7} + 23 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 13 | \( 1 - 8 T - 2 T^{2} + 164 T^{3} - 97 T^{4} - 2296 T^{5} + 956 T^{6} + 30716 T^{7} - 37243 T^{8} - 221764 T^{9} + 1015538 T^{10} - 221764 p T^{11} - 37243 p^{2} T^{12} + 30716 p^{3} T^{13} + 956 p^{4} T^{14} - 2296 p^{5} T^{15} - 97 p^{6} T^{16} + 164 p^{7} T^{17} - 2 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) |
| 17 | \( 1 - 10 T + 50 T^{2} - 108 T^{3} - 621 T^{4} + 7512 T^{5} - 39240 T^{6} + 109230 T^{7} + 90333 T^{8} - 2740564 T^{9} + 15680626 T^{10} - 2740564 p T^{11} + 90333 p^{2} T^{12} + 109230 p^{3} T^{13} - 39240 p^{4} T^{14} + 7512 p^{5} T^{15} - 621 p^{6} T^{16} - 108 p^{7} T^{17} + 50 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \) |
| 23 | \( 1 + 2 T - 43 T^{2} - 226 T^{3} + 619 T^{4} + 8068 T^{5} + 13074 T^{6} - 182772 T^{7} - 893739 T^{8} + 1802774 T^{9} + 27233383 T^{10} + 1802774 p T^{11} - 893739 p^{2} T^{12} - 182772 p^{3} T^{13} + 13074 p^{4} T^{14} + 8068 p^{5} T^{15} + 619 p^{6} T^{16} - 226 p^{7} T^{17} - 43 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) |
| 29 | \( 1 + 7 T - 34 T^{2} - 219 T^{3} + 753 T^{4} + 930 T^{5} - 24438 T^{6} - 48036 T^{7} + 10857 T^{8} + 2945899 T^{9} + 31906096 T^{10} + 2945899 p T^{11} + 10857 p^{2} T^{12} - 48036 p^{3} T^{13} - 24438 p^{4} T^{14} + 930 p^{5} T^{15} + 753 p^{6} T^{16} - 219 p^{7} T^{17} - 34 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \) |
| 31 | \( ( 1 + 9 T + 125 T^{2} + 894 T^{3} + 7009 T^{4} + 38247 T^{5} + 7009 p T^{6} + 894 p^{2} T^{7} + 125 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 37 | \( ( 1 - 6 T + 98 T^{2} - 416 T^{3} + 5953 T^{4} - 23380 T^{5} + 5953 p T^{6} - 416 p^{2} T^{7} + 98 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 41 | \( 1 - 12 T - 85 T^{2} + 980 T^{3} + 10503 T^{4} - 69872 T^{5} - 18406 p T^{6} + 2374464 T^{7} + 48595133 T^{8} - 53346940 T^{9} - 2136813723 T^{10} - 53346940 p T^{11} + 48595133 p^{2} T^{12} + 2374464 p^{3} T^{13} - 18406 p^{5} T^{14} - 69872 p^{5} T^{15} + 10503 p^{6} T^{16} + 980 p^{7} T^{17} - 85 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) |
| 43 | \( 1 - 8 T - 152 T^{2} + 884 T^{3} + 17633 T^{4} - 67756 T^{5} - 1371994 T^{6} + 2731976 T^{7} + 85754057 T^{8} - 57137584 T^{9} - 4071112762 T^{10} - 57137584 p T^{11} + 85754057 p^{2} T^{12} + 2731976 p^{3} T^{13} - 1371994 p^{4} T^{14} - 67756 p^{5} T^{15} + 17633 p^{6} T^{16} + 884 p^{7} T^{17} - 152 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) |
| 47 | \( 1 - 6 T - 46 T^{2} + 328 T^{3} + 39 T^{4} + 7184 T^{5} + 3392 p T^{6} - 2206446 T^{7} - 135091 T^{8} + 54229576 T^{9} - 121183278 T^{10} + 54229576 p T^{11} - 135091 p^{2} T^{12} - 2206446 p^{3} T^{13} + 3392 p^{5} T^{14} + 7184 p^{5} T^{15} + 39 p^{6} T^{16} + 328 p^{7} T^{17} - 46 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) |
| 53 | \( 1 - 8 T - 2 p T^{2} + 812 T^{3} + 6623 T^{4} - 41696 T^{5} - 286300 T^{6} + 1760036 T^{7} + 4817125 T^{8} - 42891780 T^{9} + 132202146 T^{10} - 42891780 p T^{11} + 4817125 p^{2} T^{12} + 1760036 p^{3} T^{13} - 286300 p^{4} T^{14} - 41696 p^{5} T^{15} + 6623 p^{6} T^{16} + 812 p^{7} T^{17} - 2 p^{9} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) |
| 59 | \( 1 + T - 2 p T^{2} + 311 T^{3} + 5831 T^{4} - 51110 T^{5} + 23624 T^{6} + 4788554 T^{7} - 15461987 T^{8} - 136511121 T^{9} + 1390161954 T^{10} - 136511121 p T^{11} - 15461987 p^{2} T^{12} + 4788554 p^{3} T^{13} + 23624 p^{4} T^{14} - 51110 p^{5} T^{15} + 5831 p^{6} T^{16} + 311 p^{7} T^{17} - 2 p^{9} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) |
| 61 | \( 1 + 7 T - 83 T^{2} - 118 T^{3} + 4022 T^{4} - 32014 T^{5} - 76597 T^{6} + 1739405 T^{7} - 4851217 T^{8} + 814892 T^{9} + 852492548 T^{10} + 814892 p T^{11} - 4851217 p^{2} T^{12} + 1739405 p^{3} T^{13} - 76597 p^{4} T^{14} - 32014 p^{5} T^{15} + 4022 p^{6} T^{16} - 118 p^{7} T^{17} - 83 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \) |
| 67 | \( 1 - 14 T + 28 T^{2} - 128 T^{3} + 6493 T^{4} - 45944 T^{5} + 583314 T^{6} - 6977406 T^{7} + 22611657 T^{8} - 258761704 T^{9} + 4586429726 T^{10} - 258761704 p T^{11} + 22611657 p^{2} T^{12} - 6977406 p^{3} T^{13} + 583314 p^{4} T^{14} - 45944 p^{5} T^{15} + 6493 p^{6} T^{16} - 128 p^{7} T^{17} + 28 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \) |
| 71 | \( 1 + 27 T + 152 T^{2} - 493 T^{3} + 20397 T^{4} + 339334 T^{5} - 286958 T^{6} - 4055352 T^{7} + 264427265 T^{8} + 982328747 T^{9} - 11469965694 T^{10} + 982328747 p T^{11} + 264427265 p^{2} T^{12} - 4055352 p^{3} T^{13} - 286958 p^{4} T^{14} + 339334 p^{5} T^{15} + 20397 p^{6} T^{16} - 493 p^{7} T^{17} + 152 p^{8} T^{18} + 27 p^{9} T^{19} + p^{10} T^{20} \) |
| 73 | \( 1 + 26 T + 250 T^{2} + 1380 T^{3} + 11199 T^{4} + 149832 T^{5} + 1122192 T^{6} - 3383358 T^{7} - 105547875 T^{8} - 462193012 T^{9} - 292651670 T^{10} - 462193012 p T^{11} - 105547875 p^{2} T^{12} - 3383358 p^{3} T^{13} + 1122192 p^{4} T^{14} + 149832 p^{5} T^{15} + 11199 p^{6} T^{16} + 1380 p^{7} T^{17} + 250 p^{8} T^{18} + 26 p^{9} T^{19} + p^{10} T^{20} \) |
| 79 | \( 1 + 23 T + 175 T^{2} + 3188 T^{3} + 57340 T^{4} + 365660 T^{5} + 4092813 T^{6} + 66231771 T^{7} + 362226147 T^{8} + 3333434920 T^{9} + 52054853960 T^{10} + 3333434920 p T^{11} + 362226147 p^{2} T^{12} + 66231771 p^{3} T^{13} + 4092813 p^{4} T^{14} + 365660 p^{5} T^{15} + 57340 p^{6} T^{16} + 3188 p^{7} T^{17} + 175 p^{8} T^{18} + 23 p^{9} T^{19} + p^{10} T^{20} \) |
| 83 | \( ( 1 - 12 T + 394 T^{2} - 3324 T^{3} + 61357 T^{4} - 384144 T^{5} + 61357 p T^{6} - 3324 p^{2} T^{7} + 394 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 89 | \( 1 + 9 T - 232 T^{2} - 1839 T^{3} + 30747 T^{4} + 178842 T^{5} - 3374208 T^{6} - 13962870 T^{7} + 304323333 T^{8} + 550905459 T^{9} - 25895228568 T^{10} + 550905459 p T^{11} + 304323333 p^{2} T^{12} - 13962870 p^{3} T^{13} - 3374208 p^{4} T^{14} + 178842 p^{5} T^{15} + 30747 p^{6} T^{16} - 1839 p^{7} T^{17} - 232 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) |
| 97 | \( ( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{5} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.62109176139026843101015426110, −3.52062454948250462706987725656, −3.47515590496716097964239144141, −3.31492392591159664152133541566, −3.30045214767654285061351965676, −3.09299939021671678124670984557, −3.01593181329600836600305933497, −2.91451809567342662777303383941, −2.75472719549048960286671918024, −2.58929031955225532426223690092, −2.57592973169373693151326182425, −2.36363389935450300142264967125, −2.27251685133949532091314124951, −1.97884053621999015866306580090, −1.93986239412817196347089364550, −1.89595407148797401177457030229, −1.88217393351276551773856321883, −1.47844804734515720309540862342, −1.45412320360186356602744331257, −1.34601421350692043109777459324, −1.18293504847763992515418138062, −1.02239810188298763216163750387, −1.01097008635882094005737397435, −0.51779915152568459489364373067, −0.18013229128337913096186245505,
0.18013229128337913096186245505, 0.51779915152568459489364373067, 1.01097008635882094005737397435, 1.02239810188298763216163750387, 1.18293504847763992515418138062, 1.34601421350692043109777459324, 1.45412320360186356602744331257, 1.47844804734515720309540862342, 1.88217393351276551773856321883, 1.89595407148797401177457030229, 1.93986239412817196347089364550, 1.97884053621999015866306580090, 2.27251685133949532091314124951, 2.36363389935450300142264967125, 2.57592973169373693151326182425, 2.58929031955225532426223690092, 2.75472719549048960286671918024, 2.91451809567342662777303383941, 3.01593181329600836600305933497, 3.09299939021671678124670984557, 3.30045214767654285061351965676, 3.31492392591159664152133541566, 3.47515590496716097964239144141, 3.52062454948250462706987725656, 3.62109176139026843101015426110
Plot not available for L-functions of degree greater than 10.