| L(s) = 1 | + 144·2-s + 1.45e3·3-s + 1.52e3·4-s − 5.25e4·5-s + 2.09e5·6-s + 2.35e5·7-s − 1.36e6·8-s + 1.59e6·9-s − 7.56e6·10-s − 8.59e6·11-s + 2.21e6·12-s − 3.79e7·13-s + 3.38e7·14-s − 7.66e7·15-s − 1.23e8·16-s + 1.66e7·17-s + 2.29e8·18-s − 2.11e8·19-s − 7.98e7·20-s + 3.43e8·21-s − 1.23e9·22-s − 1.94e8·23-s − 1.99e9·24-s + 9.18e8·25-s − 5.46e9·26-s + 1.54e9·27-s + 3.57e8·28-s + ⋯ |
| L(s) = 1 | + 1.59·2-s + 1.15·3-s + 0.185·4-s − 1.50·5-s + 1.83·6-s + 0.755·7-s − 1.84·8-s + 9-s − 2.39·10-s − 1.46·11-s + 0.214·12-s − 2.18·13-s + 1.20·14-s − 1.73·15-s − 1.84·16-s + 0.167·17-s + 1.59·18-s − 1.02·19-s − 0.279·20-s + 0.872·21-s − 2.32·22-s − 0.274·23-s − 2.13·24-s + 0.752·25-s − 3.47·26-s + 0.769·27-s + 0.140·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p^{6} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p^{6} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - 9 p^{4} T + 1201 p^{4} T^{2} - 9 p^{17} T^{3} + p^{26} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 10512 p T + 73763578 p^{2} T^{2} + 10512 p^{14} T^{3} + p^{26} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 781236 p T + 551670639946 p^{2} T^{2} + 781236 p^{14} T^{3} + p^{26} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 37982012 T + 854172190564734 T^{2} + 37982012 p^{13} T^{3} + p^{26} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 16643088 T + 4110551987575522 T^{2} - 16643088 p^{13} T^{3} + p^{26} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 211195304 T + 37306293658457814 T^{2} + 211195304 p^{13} T^{3} + p^{26} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 194997852 T + 640913258073572530 T^{2} + 194997852 p^{13} T^{3} + p^{26} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3481642548 T + 6322342576139078206 T^{2} + 3481642548 p^{13} T^{3} + p^{26} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8393797520 T + 47374586993368068990 T^{2} + 8393797520 p^{13} T^{3} + p^{26} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 22337946716 T + \)\(56\!\cdots\!86\)\( T^{2} + 22337946716 p^{13} T^{3} + p^{26} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12507403320 T + \)\(12\!\cdots\!42\)\( T^{2} + 12507403320 p^{13} T^{3} + p^{26} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 26062746328 T + \)\(32\!\cdots\!54\)\( T^{2} - 26062746328 p^{13} T^{3} + p^{26} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 29222658936 T + \)\(49\!\cdots\!30\)\( T^{2} - 29222658936 p^{13} T^{3} + p^{26} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 523911868308 T + \)\(12\!\cdots\!90\)\( T^{2} - 523911868308 p^{13} T^{3} + p^{26} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 94444581024 T + \)\(21\!\cdots\!14\)\( T^{2} + 94444581024 p^{13} T^{3} + p^{26} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 461720554252 T + \)\(31\!\cdots\!26\)\( T^{2} - 461720554252 p^{13} T^{3} + p^{26} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1237879942064 T + \)\(14\!\cdots\!30\)\( T^{2} + 1237879942064 p^{13} T^{3} + p^{26} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1961126619804 T + \)\(32\!\cdots\!26\)\( T^{2} - 1961126619804 p^{13} T^{3} + p^{26} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2407791783284 T + \)\(47\!\cdots\!62\)\( T^{2} + 2407791783284 p^{13} T^{3} + p^{26} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2622494223848 T + \)\(11\!\cdots\!42\)\( T^{2} + 2622494223848 p^{13} T^{3} + p^{26} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4489561815912 T + \)\(20\!\cdots\!90\)\( T^{2} - 4489561815912 p^{13} T^{3} + p^{26} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 520927840584 T + \)\(43\!\cdots\!50\)\( T^{2} - 520927840584 p^{13} T^{3} + p^{26} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 13027956631604 T + \)\(17\!\cdots\!86\)\( T^{2} + 13027956631604 p^{13} T^{3} + p^{26} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.65590751716431028639040140083, −14.26227205068090053689335450945, −13.35409238367345895075591483821, −12.98989220503368054268139818674, −12.19894098090498121735117302799, −12.02191579399852706233448793998, −10.80780546020790169533284303884, −9.999282118151893424843076686468, −8.966425352135372656037529158593, −8.426708799007276381686177857521, −7.49204946915743827333780400013, −7.34711936090113565869909731791, −5.34999442661610907200652220046, −5.09984952624912156157940130492, −4.00418300858214072081370990258, −3.96114119051193182764480743006, −2.79404467674201695885620491172, −2.16139889903251882746718150007, 0, 0,
2.16139889903251882746718150007, 2.79404467674201695885620491172, 3.96114119051193182764480743006, 4.00418300858214072081370990258, 5.09984952624912156157940130492, 5.34999442661610907200652220046, 7.34711936090113565869909731791, 7.49204946915743827333780400013, 8.426708799007276381686177857521, 8.966425352135372656037529158593, 9.999282118151893424843076686468, 10.80780546020790169533284303884, 12.02191579399852706233448793998, 12.19894098090498121735117302799, 12.98989220503368054268139818674, 13.35409238367345895075591483821, 14.26227205068090053689335450945, 14.65590751716431028639040140083
