L(s) = 1 | − 64·2-s − 26·3-s + 4.09e3·4-s − 1.56e4·5-s + 1.66e3·6-s + 5.38e5·7-s − 2.62e5·8-s − 1.59e6·9-s + 1.00e6·10-s − 3.99e6·11-s − 1.06e5·12-s − 2.38e7·13-s − 3.44e7·14-s + 4.06e5·15-s + 1.67e7·16-s − 1.92e8·17-s + 1.01e8·18-s + 1.66e8·19-s − 6.40e7·20-s − 1.40e7·21-s + 2.55e8·22-s − 3.66e8·23-s + 6.81e6·24-s + 2.44e8·25-s + 1.52e9·26-s + 8.28e7·27-s + 2.20e9·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.0205·3-s + 1/2·4-s − 0.447·5-s + 0.0145·6-s + 1.73·7-s − 0.353·8-s − 0.999·9-s + 0.316·10-s − 0.679·11-s − 0.0102·12-s − 1.36·13-s − 1.22·14-s + 0.00920·15-s + 1/4·16-s − 1.93·17-s + 0.706·18-s + 0.811·19-s − 0.223·20-s − 0.0356·21-s + 0.480·22-s − 0.516·23-s + 0.00728·24-s + 1/5·25-s + 0.968·26-s + 0.0411·27-s + 0.865·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p^{6} T \) |
| 5 | \( 1 + p^{6} T \) |
good | 3 | \( 1 + 26 T + p^{13} T^{2} \) |
| 7 | \( 1 - 76934 p T + p^{13} T^{2} \) |
| 11 | \( 1 + 363168 p T + p^{13} T^{2} \) |
| 13 | \( 1 + 23834446 T + p^{13} T^{2} \) |
| 17 | \( 1 + 192273222 T + p^{13} T^{2} \) |
| 19 | \( 1 - 166485740 T + p^{13} T^{2} \) |
| 23 | \( 1 + 366866946 T + p^{13} T^{2} \) |
| 29 | \( 1 - 989855670 T + p^{13} T^{2} \) |
| 31 | \( 1 + 3445048468 T + p^{13} T^{2} \) |
| 37 | \( 1 + 29429851822 T + p^{13} T^{2} \) |
| 41 | \( 1 - 7043712582 T + p^{13} T^{2} \) |
| 43 | \( 1 - 8228005214 T + p^{13} T^{2} \) |
| 47 | \( 1 - 45741859938 T + p^{13} T^{2} \) |
| 53 | \( 1 + 90591954486 T + p^{13} T^{2} \) |
| 59 | \( 1 - 126033098940 T + p^{13} T^{2} \) |
| 61 | \( 1 + 292123673038 T + p^{13} T^{2} \) |
| 67 | \( 1 - 572402067098 T + p^{13} T^{2} \) |
| 71 | \( 1 + 1284329422908 T + p^{13} T^{2} \) |
| 73 | \( 1 - 196494986594 T + p^{13} T^{2} \) |
| 79 | \( 1 - 3776797097000 T + p^{13} T^{2} \) |
| 83 | \( 1 + 4556844205746 T + p^{13} T^{2} \) |
| 89 | \( 1 - 3748393684890 T + p^{13} T^{2} \) |
| 97 | \( 1 + 2743981383742 T + p^{13} T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.28236805244698199893653210928, −15.49537667826023387680416424718, −14.23811847267004920085271440081, −11.86099267144663710576334497312, −10.84110787001852964696272359890, −8.710718539615435959952989541319, −7.51821927702662037354979076128, −5.00782424948703379719266171494, −2.21969145174946915955485347354, 0,
2.21969145174946915955485347354, 5.00782424948703379719266171494, 7.51821927702662037354979076128, 8.710718539615435959952989541319, 10.84110787001852964696272359890, 11.86099267144663710576334497312, 14.23811847267004920085271440081, 15.49537667826023387680416424718, 17.28236805244698199893653210928