| L(s) = 1 | + 16·2-s + 62·3-s + 256·4-s + 766·5-s + 992·6-s + 2.40e3·7-s + 4.09e3·8-s − 2.71e3·9-s + 1.22e4·10-s + 1.58e4·12-s − 3.89e4·13-s + 3.84e4·14-s + 4.74e4·15-s + 6.55e4·16-s − 4.34e4·18-s − 1.61e5·19-s + 1.96e5·20-s + 1.48e5·21-s + 3.58e5·23-s + 2.53e5·24-s + 1.96e5·25-s − 6.23e5·26-s − 5.75e5·27-s + 6.14e5·28-s + 7.59e5·30-s + 1.04e6·32-s + 1.83e6·35-s + ⋯ |
| L(s) = 1 | + 2-s + 0.765·3-s + 4-s + 1.22·5-s + 0.765·6-s + 7-s + 8-s − 0.414·9-s + 1.22·10-s + 0.765·12-s − 1.36·13-s + 14-s + 0.938·15-s + 16-s − 0.414·18-s − 1.24·19-s + 1.22·20-s + 0.765·21-s + 1.27·23-s + 0.765·24-s + 0.502·25-s − 1.36·26-s − 1.08·27-s + 28-s + 0.938·30-s + 32-s + 1.22·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(5.785940967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.785940967\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{4} T \) |
| 7 | \( 1 - p^{4} T \) |
| good | 3 | \( 1 - 62 T + p^{8} T^{2} \) |
| 5 | \( 1 - 766 T + p^{8} T^{2} \) |
| 11 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 13 | \( 1 + 38978 T + p^{8} T^{2} \) |
| 17 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 19 | \( 1 + 161858 T + p^{8} T^{2} \) |
| 23 | \( 1 - 358082 T + p^{8} T^{2} \) |
| 29 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 31 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 37 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 41 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 43 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 47 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 53 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 59 | \( 1 - 22957822 T + p^{8} T^{2} \) |
| 61 | \( 1 + 26625218 T + p^{8} T^{2} \) |
| 67 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 71 | \( 1 + 46751038 T + p^{8} T^{2} \) |
| 73 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 79 | \( 1 + 58381438 T + p^{8} T^{2} \) |
| 83 | \( 1 + 77480258 T + p^{8} T^{2} \) |
| 89 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 97 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
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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
−13.74168986895726530515566898096, −12.75604625425718619504707387921, −11.39524997824341770934129761073, −10.16279317330891742242918733222, −8.726181637610927335618826706564, −7.30596877661658962010608481232, −5.77508863671967657414185612013, −4.65267654783670375764485658090, −2.73007466738412520546643980057, −1.84909782938458667028205801630,
1.84909782938458667028205801630, 2.73007466738412520546643980057, 4.65267654783670375764485658090, 5.77508863671967657414185612013, 7.30596877661658962010608481232, 8.726181637610927335618826706564, 10.16279317330891742242918733222, 11.39524997824341770934129761073, 12.75604625425718619504707387921, 13.74168986895726530515566898096
