| L(s) = 1 | − 729·3-s − 2.24e4·5-s − 1.81e5·7-s + 5.31e5·9-s + 9.26e6·11-s + 2.99e5·13-s + 1.63e7·15-s + 1.32e7·17-s − 8.70e7·19-s + 1.32e8·21-s + 9.92e8·23-s − 7.14e8·25-s − 3.87e8·27-s − 1.97e9·29-s + 7.52e9·31-s − 6.75e9·33-s + 4.07e9·35-s − 7.06e9·37-s − 2.17e8·39-s − 3.55e10·41-s + 3.89e9·43-s − 1.19e10·45-s − 3.16e10·47-s − 6.40e10·49-s − 9.65e9·51-s − 7.89e10·53-s − 2.08e11·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.643·5-s − 0.582·7-s + 1/3·9-s + 1.57·11-s + 0.0171·13-s + 0.371·15-s + 0.133·17-s − 0.424·19-s + 0.336·21-s + 1.39·23-s − 0.585·25-s − 0.192·27-s − 0.616·29-s + 1.52·31-s − 0.910·33-s + 0.374·35-s − 0.452·37-s − 0.00992·39-s − 1.16·41-s + 0.0938·43-s − 0.214·45-s − 0.428·47-s − 0.660·49-s − 0.0768·51-s − 0.489·53-s − 1.01·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + p^{6} T \) |
| good | 5 | \( 1 + 4498 p T + p^{13} T^{2} \) |
| 7 | \( 1 + 25896 p T + p^{13} T^{2} \) |
| 11 | \( 1 - 841948 p T + p^{13} T^{2} \) |
| 13 | \( 1 - 299038 T + p^{13} T^{2} \) |
| 17 | \( 1 - 13249394 T + p^{13} T^{2} \) |
| 19 | \( 1 + 87090068 T + p^{13} T^{2} \) |
| 23 | \( 1 - 992273096 T + p^{13} T^{2} \) |
| 29 | \( 1 + 1975365762 T + p^{13} T^{2} \) |
| 31 | \( 1 - 7521761680 T + p^{13} T^{2} \) |
| 37 | \( 1 + 7061279370 T + p^{13} T^{2} \) |
| 41 | \( 1 + 35578163478 T + p^{13} T^{2} \) |
| 43 | \( 1 - 3892318868 T + p^{13} T^{2} \) |
| 47 | \( 1 + 31686836880 T + p^{13} T^{2} \) |
| 53 | \( 1 + 78937081610 T + p^{13} T^{2} \) |
| 59 | \( 1 + 287098824604 T + p^{13} T^{2} \) |
| 61 | \( 1 - 620132700142 T + p^{13} T^{2} \) |
| 67 | \( 1 + 970682538788 T + p^{13} T^{2} \) |
| 71 | \( 1 + 1086445967336 T + p^{13} T^{2} \) |
| 73 | \( 1 + 2089728265814 T + p^{13} T^{2} \) |
| 79 | \( 1 + 3777056177632 T + p^{13} T^{2} \) |
| 83 | \( 1 - 3237672550444 T + p^{13} T^{2} \) |
| 89 | \( 1 + 6361576141254 T + p^{13} T^{2} \) |
| 97 | \( 1 - 11827133198882 T + p^{13} T^{2} \) |
| 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
−12.06657171919292928701220346194, −11.34024650552893989044106859326, −9.932733229217994479798568672180, −8.722794921456575221945978482577, −7.12800516185429241179114129374, −6.17260781669676898651878848764, −4.52787464018331625405750035690, −3.35394776246154016868030011001, −1.33052831029954071358858976751, 0,
1.33052831029954071358858976751, 3.35394776246154016868030011001, 4.52787464018331625405750035690, 6.17260781669676898651878848764, 7.12800516185429241179114129374, 8.722794921456575221945978482577, 9.932733229217994479798568672180, 11.34024650552893989044106859326, 12.06657171919292928701220346194
