| L(s) = 1 | + 18·2-s − 27·3-s + 196·4-s − 125·5-s − 486·6-s + 343·7-s + 1.22e3·8-s + 729·9-s − 2.25e3·10-s − 8.01e3·11-s − 5.29e3·12-s − 1.78e3·13-s + 6.17e3·14-s + 3.37e3·15-s − 3.05e3·16-s + 8.35e3·17-s + 1.31e4·18-s − 5.88e3·19-s − 2.45e4·20-s − 9.26e3·21-s − 1.44e5·22-s − 7.77e4·23-s − 3.30e4·24-s + 1.56e4·25-s − 3.21e4·26-s − 1.96e4·27-s + 6.72e4·28-s + ⋯ |
| L(s) = 1 | + 1.59·2-s − 0.577·3-s + 1.53·4-s − 0.447·5-s − 0.918·6-s + 0.377·7-s + 0.845·8-s + 1/3·9-s − 0.711·10-s − 1.81·11-s − 0.884·12-s − 0.225·13-s + 0.601·14-s + 0.258·15-s − 0.186·16-s + 0.412·17-s + 0.530·18-s − 0.196·19-s − 0.684·20-s − 0.218·21-s − 2.88·22-s − 1.33·23-s − 0.487·24-s + 1/5·25-s − 0.358·26-s − 0.192·27-s + 0.578·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p^{3} T \) |
| 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
−12.16254937676992421354150343841, −11.15287598738945318265208731490, −10.22686703556795109929091112336, −8.169435791781351683343630055878, −7.02145039367259378666979812259, −5.60527185895786294267640231270, −4.96209616146296597357367104490, −3.69991664601892506108962381353, −2.27186155217282885274184668286, 0,
2.27186155217282885274184668286, 3.69991664601892506108962381353, 4.96209616146296597357367104490, 5.60527185895786294267640231270, 7.02145039367259378666979812259, 8.169435791781351683343630055878, 10.22686703556795109929091112336, 11.15287598738945318265208731490, 12.16254937676992421354150343841
