L(s) = 1 | + 3·2-s + 3·3-s + 3·4-s − 3·5-s + 9·6-s − 3·7-s − 2·8-s + 3·9-s − 9·10-s + 9·12-s − 6·13-s − 9·14-s − 9·15-s − 9·16-s − 3·17-s + 9·18-s − 6·19-s − 9·20-s − 9·21-s + 3·23-s − 6·24-s + 18·25-s − 18·26-s − 2·27-s − 9·28-s + 24·29-s − 27·30-s + ⋯ |
L(s) = 1 | + 2.12·2-s + 1.73·3-s + 3/2·4-s − 1.34·5-s + 3.67·6-s − 1.13·7-s − 0.707·8-s + 9-s − 2.84·10-s + 2.59·12-s − 1.66·13-s − 2.40·14-s − 2.32·15-s − 9/4·16-s − 0.727·17-s + 2.12·18-s − 1.37·19-s − 2.01·20-s − 1.96·21-s + 0.625·23-s − 1.22·24-s + 18/5·25-s − 3.53·26-s − 0.384·27-s − 1.70·28-s + 4.45·29-s − 4.92·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2815263942\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2815263942\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( ( 1 - T + T^{2} )^{3} \) |
| 3 | \( ( 1 - T + T^{2} )^{3} \) |
| 7 | \( 1 + 3 T + 6 T^{2} + 5 T^{3} + 6 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | \( ( 1 - T + T^{2} )^{3} \) |
good | 5 | \( ( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} )^{3} \) |
| 11 | \( 1 - 18 T^{2} - 12 T^{3} + 126 T^{4} + 108 T^{5} - 866 T^{6} + 108 p T^{7} + 126 p^{2} T^{8} - 12 p^{3} T^{9} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( ( 1 + 3 T + 15 T^{2} + 56 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 17 | \( 1 + 3 T - 6 T^{2} + 93 T^{3} + 42 T^{4} - 1023 T^{5} + 3976 T^{6} - 1023 p T^{7} + 42 p^{2} T^{8} + 93 p^{3} T^{9} - 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 19 | \( 1 + 6 T - 15 T^{2} - 54 T^{3} + 552 T^{4} - 282 T^{5} - 16297 T^{6} - 282 p T^{7} + 552 p^{2} T^{8} - 54 p^{3} T^{9} - 15 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 12 T + 108 T^{2} - 648 T^{3} + 108 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 18 T^{2} + 428 T^{3} - 234 T^{4} - 3852 T^{5} + 95334 T^{6} - 3852 p T^{7} - 234 p^{2} T^{8} + 428 p^{3} T^{9} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( 1 - 12 T + 75 T^{2} - 228 T^{3} - 1392 T^{4} + 24144 T^{5} - 181987 T^{6} + 24144 p T^{7} - 1392 p^{2} T^{8} - 228 p^{3} T^{9} + 75 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 18 T + 213 T^{2} + 1604 T^{3} + 213 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( ( 1 + 69 T^{2} + 48 T^{3} + 69 p T^{4} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 3 T - 60 T^{2} + 637 T^{3} + 252 T^{4} - 16287 T^{5} + 142082 T^{6} - 16287 p T^{7} + 252 p^{2} T^{8} + 637 p^{3} T^{9} - 60 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 - 3 T - 93 T^{2} + 308 T^{3} + 4077 T^{4} - 7305 T^{5} - 184690 T^{6} - 7305 p T^{7} + 4077 p^{2} T^{8} + 308 p^{3} T^{9} - 93 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 - 12 T - 42 T^{2} + 240 T^{3} + 8790 T^{4} - 8292 T^{5} - 603974 T^{6} - 8292 p T^{7} + 8790 p^{2} T^{8} + 240 p^{3} T^{9} - 42 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( ( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{3} \) |
| 67 | \( 1 - 21 T + 168 T^{2} - 897 T^{3} + 5460 T^{4} + 13839 T^{5} - 524842 T^{6} + 13839 p T^{7} + 5460 p^{2} T^{8} - 897 p^{3} T^{9} + 168 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 3 T + 81 T^{2} - 300 T^{3} + 81 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 + 15 T - 54 T^{2} - 307 T^{3} + 23784 T^{4} + 95559 T^{5} - 1058868 T^{6} + 95559 p T^{7} + 23784 p^{2} T^{8} - 307 p^{3} T^{9} - 54 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 - 12 T - 6 T^{2} - 144 T^{3} + 714 T^{4} + 85308 T^{5} - 1004542 T^{6} + 85308 p T^{7} + 714 p^{2} T^{8} - 144 p^{3} T^{9} - 6 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( ( 1 + 12 T + 282 T^{2} + 1990 T^{3} + 282 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 + 12 T - 81 T^{2} - 396 T^{3} + 12024 T^{4} - 37248 T^{5} - 1840007 T^{6} - 37248 p T^{7} + 12024 p^{2} T^{8} - 396 p^{3} T^{9} - 81 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( ( 1 + 48 T^{2} + 108 T^{3} + 48 p T^{4} + p^{3} T^{6} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.15997556293369945049991452662, −5.01973266763544188285357758943, −4.86464688344343991132537336350, −4.70970745490661300677178128648, −4.59150282303395932057951645637, −4.52264858475533961490706147186, −4.50019237711015107665076555007, −4.02239022172035898735972092187, −4.00936698369498307461884093918, −3.81665551182620516028775195362, −3.56222591461921974171309353028, −3.44216289169571614073568948012, −3.32028521503499150129832537339, −3.03593217699854680678221819539, −2.89157347783311389512588510559, −2.74532695135645841138657585530, −2.65755195892137485364228377719, −2.63909438833366448904268279161, −2.40677028833342114359631031350, −2.06951049550122921765105534901, −1.63133496660043055626531320438, −1.34310429369258863994140254302, −1.10986971790949455538385014326, −0.59308165956348605226966844169, −0.05915110394535300484896955737,
0.05915110394535300484896955737, 0.59308165956348605226966844169, 1.10986971790949455538385014326, 1.34310429369258863994140254302, 1.63133496660043055626531320438, 2.06951049550122921765105534901, 2.40677028833342114359631031350, 2.63909438833366448904268279161, 2.65755195892137485364228377719, 2.74532695135645841138657585530, 2.89157347783311389512588510559, 3.03593217699854680678221819539, 3.32028521503499150129832537339, 3.44216289169571614073568948012, 3.56222591461921974171309353028, 3.81665551182620516028775195362, 4.00936698369498307461884093918, 4.02239022172035898735972092187, 4.50019237711015107665076555007, 4.52264858475533961490706147186, 4.59150282303395932057951645637, 4.70970745490661300677178128648, 4.86464688344343991132537336350, 5.01973266763544188285357758943, 5.15997556293369945049991452662
Plot not available for L-functions of degree greater than 10.