| L(s) = 1 | + 8·2-s + 27·3-s + 64·4-s − 10·5-s + 216·6-s − 343·7-s + 512·8-s + 729·9-s − 80·10-s + 1.50e3·11-s + 1.72e3·12-s + 2.19e3·13-s − 2.74e3·14-s − 270·15-s + 4.09e3·16-s − 1.04e3·17-s + 5.83e3·18-s − 9.06e3·19-s − 640·20-s − 9.26e3·21-s + 1.20e4·22-s − 9.89e4·23-s + 1.38e4·24-s − 7.80e4·25-s + 1.75e4·26-s + 1.96e4·27-s − 2.19e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.0357·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.0252·10-s + 0.341·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.0206·15-s + 1/4·16-s − 0.0514·17-s + 0.235·18-s − 0.303·19-s − 0.0178·20-s − 0.218·21-s + 0.241·22-s − 1.69·23-s + 0.204·24-s − 0.998·25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{3} T \) |
| 3 | \( 1 - p^{3} T \) |
| 7 | \( 1 + p^{3} T \) |
| 13 | \( 1 - p^{3} T \) |
| good | 5 | \( 1 + 2 p T + p^{7} T^{2} \) |
| 11 | \( 1 - 1508 T + p^{7} T^{2} \) |
| 17 | \( 1 + 1042 T + p^{7} T^{2} \) |
| 19 | \( 1 + 9068 T + p^{7} T^{2} \) |
| 23 | \( 1 + 98988 T + p^{7} T^{2} \) |
| 29 | \( 1 + 213642 T + p^{7} T^{2} \) |
| 31 | \( 1 + 22048 T + p^{7} T^{2} \) |
| 37 | \( 1 - 418246 T + p^{7} T^{2} \) |
| 41 | \( 1 + 76414 T + p^{7} T^{2} \) |
| 43 | \( 1 + 177524 T + p^{7} T^{2} \) |
| 47 | \( 1 - 631916 T + p^{7} T^{2} \) |
| 53 | \( 1 + 982354 T + p^{7} T^{2} \) |
| 59 | \( 1 + 596384 T + p^{7} T^{2} \) |
| 61 | \( 1 - 1863406 T + p^{7} T^{2} \) |
| 67 | \( 1 + 1845652 T + p^{7} T^{2} \) |
| 71 | \( 1 - 1632280 T + p^{7} T^{2} \) |
| 73 | \( 1 - 216650 T + p^{7} T^{2} \) |
| 79 | \( 1 + 6460168 T + p^{7} T^{2} \) |
| 83 | \( 1 - 3592152 T + p^{7} T^{2} \) |
| 89 | \( 1 + 9482662 T + p^{7} T^{2} \) |
| 97 | \( 1 - 840226 T + p^{7} T^{2} \) |
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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
−9.323215567299887563768310128728, −8.194205921707959177733702930532, −7.43956525317654958554676863649, −6.34969379181831723846537577133, −5.62869530701833990314981776336, −4.20721176332782513868914523472, −3.69569971310356149828961154530, −2.49219681664845626921846534963, −1.57837734094700720670726401602, 0,
1.57837734094700720670726401602, 2.49219681664845626921846534963, 3.69569971310356149828961154530, 4.20721176332782513868914523472, 5.62869530701833990314981776336, 6.34969379181831723846537577133, 7.43956525317654958554676863649, 8.194205921707959177733702930532, 9.323215567299887563768310128728
