L(s) = 1 | − 64·2-s + 1.62e3·3-s + 4.09e3·4-s − 3.64e4·5-s − 1.04e5·6-s − 1.17e5·7-s − 2.62e5·8-s + 1.04e6·9-s + 2.32e6·10-s + 2.60e6·11-s + 6.66e6·12-s − 1.26e7·13-s + 7.52e6·14-s − 5.91e7·15-s + 1.67e7·16-s − 1.30e8·17-s − 6.71e7·18-s − 2.49e8·19-s − 1.49e8·20-s − 1.91e8·21-s − 1.66e8·22-s + 4.89e8·23-s − 4.26e8·24-s + 1.04e8·25-s + 8.07e8·26-s − 8.85e8·27-s − 4.81e8·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.28·3-s + 1/2·4-s − 1.04·5-s − 0.910·6-s − 0.377·7-s − 0.353·8-s + 0.658·9-s + 0.736·10-s + 0.443·11-s + 0.643·12-s − 0.725·13-s + 0.267·14-s − 1.34·15-s + 1/4·16-s − 1.31·17-s − 0.465·18-s − 1.21·19-s − 0.520·20-s − 0.486·21-s − 0.313·22-s + 0.688·23-s − 0.455·24-s + 0.0854·25-s + 0.512·26-s − 0.440·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(7)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{15}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + p^{6} T \) |
| 7 | \( 1 + p^{6} T \) |
good | 3 | \( 1 - 542 p T + p^{13} T^{2} \) |
| 5 | \( 1 + 1456 p^{2} T + p^{13} T^{2} \) |
| 11 | \( 1 - 2605288 T + p^{13} T^{2} \) |
| 13 | \( 1 + 12624468 T + p^{13} T^{2} \) |
| 17 | \( 1 + 130752362 T + p^{13} T^{2} \) |
| 19 | \( 1 + 249436042 T + p^{13} T^{2} \) |
| 23 | \( 1 - 489054160 T + p^{13} T^{2} \) |
| 29 | \( 1 + 112115926 T + p^{13} T^{2} \) |
| 31 | \( 1 + 9103068684 T + p^{13} T^{2} \) |
| 37 | \( 1 - 18308169938 T + p^{13} T^{2} \) |
| 41 | \( 1 - 13082373606 T + p^{13} T^{2} \) |
| 43 | \( 1 + 67123460032 T + p^{13} T^{2} \) |
| 47 | \( 1 - 105239980284 T + p^{13} T^{2} \) |
| 53 | \( 1 + 25221720042 T + p^{13} T^{2} \) |
| 59 | \( 1 + 276774602098 T + p^{13} T^{2} \) |
| 61 | \( 1 - 759388645560 T + p^{13} T^{2} \) |
| 67 | \( 1 - 1039664575708 T + p^{13} T^{2} \) |
| 71 | \( 1 - 1817086195456 T + p^{13} T^{2} \) |
| 73 | \( 1 - 400342248850 T + p^{13} T^{2} \) |
| 79 | \( 1 + 3597798513336 T + p^{13} T^{2} \) |
| 83 | \( 1 + 1309030493954 T + p^{13} T^{2} \) |
| 89 | \( 1 - 1653288354570 T + p^{13} T^{2} \) |
| 97 | \( 1 + 12736909073690 T + p^{13} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.50717519594399966232332399858, −14.67046229346699721168475761155, −12.87601627344683366244344785736, −11.19695770861436642835502524317, −9.352290978282140077635738633780, −8.333209898535310523708836747723, −7.03969134084599732022202354473, −3.89084055628901776798802155104, −2.32374369602858636946108814757, 0,
2.32374369602858636946108814757, 3.89084055628901776798802155104, 7.03969134084599732022202354473, 8.333209898535310523708836747723, 9.352290978282140077635738633780, 11.19695770861436642835502524317, 12.87601627344683366244344785736, 14.67046229346699721168475761155, 15.50717519594399966232332399858