| L(s) = 1 | − 8·2-s − 27·3-s + 64·4-s + 182·5-s + 216·6-s + 343·7-s − 512·8-s + 729·9-s − 1.45e3·10-s + 1.33e3·11-s − 1.72e3·12-s + 1.37e4·13-s − 2.74e3·14-s − 4.91e3·15-s + 4.09e3·16-s + 1.47e4·17-s − 5.83e3·18-s + 3.45e4·19-s + 1.16e4·20-s − 9.26e3·21-s − 1.06e4·22-s + 4.98e4·23-s + 1.38e4·24-s − 4.50e4·25-s − 1.10e5·26-s − 1.96e4·27-s + 2.19e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.651·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.460·10-s + 0.301·11-s − 0.288·12-s + 1.73·13-s − 0.267·14-s − 0.375·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 1.15·19-s + 0.325·20-s − 0.218·21-s − 0.213·22-s + 0.853·23-s + 0.204·24-s − 0.576·25-s − 1.22·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.255891734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.255891734\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - p^{3} T \) |
| 11 | \( 1 - p^{3} T \) |
| good | 5 | \( 1 - 182 T + p^{7} T^{2} \) |
| 13 | \( 1 - 13758 T + p^{7} T^{2} \) |
| 17 | \( 1 - 14738 T + p^{7} T^{2} \) |
| 19 | \( 1 - 34516 T + p^{7} T^{2} \) |
| 23 | \( 1 - 49816 T + p^{7} T^{2} \) |
| 29 | \( 1 - 141174 T + p^{7} T^{2} \) |
| 31 | \( 1 + 86112 T + p^{7} T^{2} \) |
| 37 | \( 1 - 535038 T + p^{7} T^{2} \) |
| 41 | \( 1 - 708282 T + p^{7} T^{2} \) |
| 43 | \( 1 - 162292 T + p^{7} T^{2} \) |
| 47 | \( 1 - 397296 T + p^{7} T^{2} \) |
| 53 | \( 1 - 224798 T + p^{7} T^{2} \) |
| 59 | \( 1 - 694668 T + p^{7} T^{2} \) |
| 61 | \( 1 - 50110 T + p^{7} T^{2} \) |
| 67 | \( 1 + 3228404 T + p^{7} T^{2} \) |
| 71 | \( 1 + 1871496 T + p^{7} T^{2} \) |
| 73 | \( 1 + 937542 T + p^{7} T^{2} \) |
| 79 | \( 1 - 1266032 T + p^{7} T^{2} \) |
| 83 | \( 1 + 2545004 T + p^{7} T^{2} \) |
| 89 | \( 1 + 9909830 T + p^{7} T^{2} \) |
| 97 | \( 1 + 14010830 T + p^{7} T^{2} \) |
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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
−9.829405920894000291691912827216, −9.133856144548257560843608906051, −8.135011519794218559051469688727, −7.19682608063025506691719511870, −6.07322749124437834456808094019, −5.59004586298853140879180687172, −4.14534935229603494998595425631, −2.84316919769323409149101251497, −1.36196405662108944880561820494, −0.922494077665475583469570564492,
0.922494077665475583469570564492, 1.36196405662108944880561820494, 2.84316919769323409149101251497, 4.14534935229603494998595425631, 5.59004586298853140879180687172, 6.07322749124437834456808094019, 7.19682608063025506691719511870, 8.135011519794218559051469688727, 9.133856144548257560843608906051, 9.829405920894000291691912827216
