| L(s) = 1 | + 64·2-s − 729·3-s + 4.09e3·4-s + 5.46e4·5-s − 4.66e4·6-s + 1.76e5·7-s + 2.62e5·8-s + 5.31e5·9-s + 3.49e6·10-s + 6.61e6·11-s − 2.98e6·12-s − 2.40e7·13-s + 1.12e7·14-s − 3.98e7·15-s + 1.67e7·16-s − 1.54e8·17-s + 3.40e7·18-s + 1.90e8·19-s + 2.23e8·20-s − 1.28e8·21-s + 4.23e8·22-s − 3.52e8·23-s − 1.91e8·24-s + 1.76e9·25-s − 1.53e9·26-s − 3.87e8·27-s + 7.22e8·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.56·5-s − 0.408·6-s + 0.566·7-s + 0.353·8-s + 1/3·9-s + 1.10·10-s + 1.12·11-s − 0.288·12-s − 1.38·13-s + 0.400·14-s − 0.903·15-s + 1/4·16-s − 1.55·17-s + 0.235·18-s + 0.926·19-s + 0.782·20-s − 0.327·21-s + 0.795·22-s − 0.497·23-s − 0.204·24-s + 1.44·25-s − 0.976·26-s − 0.192·27-s + 0.283·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(2.571992364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.571992364\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{6} T \) |
| 3 | \( 1 + p^{6} T \) |
| good | 5 | \( 1 - 54654 T + p^{13} T^{2} \) |
| 7 | \( 1 - 176336 T + p^{13} T^{2} \) |
| 11 | \( 1 - 6612420 T + p^{13} T^{2} \) |
| 13 | \( 1 + 24028978 T + p^{13} T^{2} \) |
| 17 | \( 1 + 154665054 T + p^{13} T^{2} \) |
| 19 | \( 1 - 190034876 T + p^{13} T^{2} \) |
| 23 | \( 1 + 352957800 T + p^{13} T^{2} \) |
| 29 | \( 1 + 2804086266 T + p^{13} T^{2} \) |
| 31 | \( 1 - 2763661208 T + p^{13} T^{2} \) |
| 37 | \( 1 - 20030257622 T + p^{13} T^{2} \) |
| 41 | \( 1 + 39624547206 T + p^{13} T^{2} \) |
| 43 | \( 1 + 81486174844 T + p^{13} T^{2} \) |
| 47 | \( 1 + 34136017440 T + p^{13} T^{2} \) |
| 53 | \( 1 + 21810829986 T + p^{13} T^{2} \) |
| 59 | \( 1 - 229219661220 T + p^{13} T^{2} \) |
| 61 | \( 1 - 9799736750 T + p^{13} T^{2} \) |
| 67 | \( 1 - 789042707996 T + p^{13} T^{2} \) |
| 71 | \( 1 + 369504705240 T + p^{13} T^{2} \) |
| 73 | \( 1 + 693077725078 T + p^{13} T^{2} \) |
| 79 | \( 1 - 2231309995208 T + p^{13} T^{2} \) |
| 83 | \( 1 - 2084328707772 T + p^{13} T^{2} \) |
| 89 | \( 1 - 2221961096538 T + p^{13} T^{2} \) |
| 97 | \( 1 - 10268379896642 T + p^{13} T^{2} \) |
| 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
−20.16538493098578525289559102984, −17.85344691463666233585676770272, −16.89588426415504340344338865514, −14.69767803193811730583335758354, −13.39014379752963156037473788113, −11.63420742543436054976583208863, −9.738638429771938790190229427093, −6.55835356163152473822760028891, −4.97443916209920337277647780904, −1.89534149440807855302493492403,
1.89534149440807855302493492403, 4.97443916209920337277647780904, 6.55835356163152473822760028891, 9.738638429771938790190229427093, 11.63420742543436054976583208863, 13.39014379752963156037473788113, 14.69767803193811730583335758354, 16.89588426415504340344338865514, 17.85344691463666233585676770272, 20.16538493098578525289559102984
