| L(s) = 1 | − 18·2-s + 196·4-s + 125·5-s + 343·7-s − 1.22e3·8-s − 2.25e3·10-s + 8.01e3·11-s − 1.78e3·13-s − 6.17e3·14-s − 3.05e3·16-s − 8.35e3·17-s − 5.88e3·19-s + 2.45e4·20-s − 1.44e5·22-s + 7.77e4·23-s + 1.56e4·25-s + 3.21e4·26-s + 6.72e4·28-s − 1.55e5·29-s − 3.10e5·31-s + 2.11e5·32-s + 1.50e5·34-s + 4.28e4·35-s − 4.33e5·37-s + 1.05e5·38-s − 1.53e5·40-s − 3.57e5·41-s + ⋯ |
| L(s) = 1 | − 1.59·2-s + 1.53·4-s + 0.447·5-s + 0.377·7-s − 0.845·8-s − 0.711·10-s + 1.81·11-s − 0.225·13-s − 0.601·14-s − 0.186·16-s − 0.412·17-s − 0.196·19-s + 0.684·20-s − 2.88·22-s + 1.33·23-s + 1/5·25-s + 0.358·26-s + 0.578·28-s − 1.18·29-s − 1.86·31-s + 1.14·32-s + 0.656·34-s + 0.169·35-s − 1.40·37-s + 0.313·38-s − 0.377·40-s − 0.811·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - p^{3} T \) |
| 7 | \( 1 - p^{3} T \) |
| good | 2 | \( 1 + 9 p T + p^{7} T^{2} \) |
| 11 | \( 1 - 8016 T + p^{7} T^{2} \) |
| 13 | \( 1 + 1786 T + p^{7} T^{2} \) |
| 17 | \( 1 + 8358 T + p^{7} T^{2} \) |
| 19 | \( 1 + 5884 T + p^{7} T^{2} \) |
| 23 | \( 1 - 77700 T + p^{7} T^{2} \) |
| 29 | \( 1 + 155742 T + p^{7} T^{2} \) |
| 31 | \( 1 + 10000 p T + p^{7} T^{2} \) |
| 37 | \( 1 + 433618 T + p^{7} T^{2} \) |
| 41 | \( 1 + 357942 T + p^{7} T^{2} \) |
| 43 | \( 1 + 724492 T + p^{7} T^{2} \) |
| 47 | \( 1 + 175320 T + p^{7} T^{2} \) |
| 53 | \( 1 + 132198 T + p^{7} T^{2} \) |
| 59 | \( 1 + 44892 p T + p^{7} T^{2} \) |
| 61 | \( 1 - 835478 T + p^{7} T^{2} \) |
| 67 | \( 1 - 3486308 T + p^{7} T^{2} \) |
| 71 | \( 1 - 2872260 T + p^{7} T^{2} \) |
| 73 | \( 1 - 5951882 T + p^{7} T^{2} \) |
| 79 | \( 1 + 1680904 T + p^{7} T^{2} \) |
| 83 | \( 1 + 3577524 T + p^{7} T^{2} \) |
| 89 | \( 1 - 6254826 T + p^{7} T^{2} \) |
| 97 | \( 1 + 5257054 T + p^{7} T^{2} \) |
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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
−9.617048099366130674806335954011, −9.151661369503961962117148145250, −8.405675446409159131043949560336, −7.13188360919454764319842406104, −6.61636499852554801866918111058, −5.13646523835169714366678375839, −3.64173694804000630406893618966, −1.94298647854702915529021799917, −1.33202535904229649704791322438, 0,
1.33202535904229649704791322438, 1.94298647854702915529021799917, 3.64173694804000630406893618966, 5.13646523835169714366678375839, 6.61636499852554801866918111058, 7.13188360919454764319842406104, 8.405675446409159131043949560336, 9.151661369503961962117148145250, 9.617048099366130674806335954011
