| L(s) = 1 | − 6·2-s − 92·4-s − 390·5-s − 64·7-s + 1.32e3·8-s + 2.34e3·10-s + 948·11-s − 5.09e3·13-s + 384·14-s + 3.85e3·16-s − 2.83e4·17-s − 8.62e3·19-s + 3.58e4·20-s − 5.68e3·22-s + 1.52e4·23-s + 7.39e4·25-s + 3.05e4·26-s + 5.88e3·28-s − 3.65e4·29-s − 2.76e5·31-s − 1.92e5·32-s + 1.70e5·34-s + 2.49e4·35-s + 2.68e5·37-s + 5.17e4·38-s − 5.14e5·40-s + 6.29e5·41-s + ⋯ |
| L(s) = 1 | − 0.530·2-s − 0.718·4-s − 1.39·5-s − 0.0705·7-s + 0.911·8-s + 0.739·10-s + 0.214·11-s − 0.643·13-s + 0.0374·14-s + 0.235·16-s − 1.40·17-s − 0.288·19-s + 1.00·20-s − 0.113·22-s + 0.262·23-s + 0.946·25-s + 0.341·26-s + 0.0506·28-s − 0.277·29-s − 1.66·31-s − 1.03·32-s + 0.743·34-s + 0.0984·35-s + 0.871·37-s + 0.152·38-s − 1.27·40-s + 1.42·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 2 | \( 1 + 3 p T + p^{7} T^{2} \) |
| 5 | \( 1 + 78 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 64 T + p^{7} T^{2} \) |
| 11 | \( 1 - 948 T + p^{7} T^{2} \) |
| 13 | \( 1 + 5098 T + p^{7} T^{2} \) |
| 17 | \( 1 + 28386 T + p^{7} T^{2} \) |
| 19 | \( 1 + 8620 T + p^{7} T^{2} \) |
| 23 | \( 1 - 15288 T + p^{7} T^{2} \) |
| 29 | \( 1 + 36510 T + p^{7} T^{2} \) |
| 31 | \( 1 + 276808 T + p^{7} T^{2} \) |
| 37 | \( 1 - 268526 T + p^{7} T^{2} \) |
| 41 | \( 1 - 629718 T + p^{7} T^{2} \) |
| 43 | \( 1 - 685772 T + p^{7} T^{2} \) |
| 47 | \( 1 + 583296 T + p^{7} T^{2} \) |
| 53 | \( 1 - 428058 T + p^{7} T^{2} \) |
| 59 | \( 1 + 1306380 T + p^{7} T^{2} \) |
| 61 | \( 1 - 300662 T + p^{7} T^{2} \) |
| 67 | \( 1 + 507244 T + p^{7} T^{2} \) |
| 71 | \( 1 + 5560632 T + p^{7} T^{2} \) |
| 73 | \( 1 - 1369082 T + p^{7} T^{2} \) |
| 79 | \( 1 + 6913720 T + p^{7} T^{2} \) |
| 83 | \( 1 - 4376748 T + p^{7} T^{2} \) |
| 89 | \( 1 - 8528310 T + p^{7} T^{2} \) |
| 97 | \( 1 + 8826814 T + p^{7} T^{2} \) |
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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
−19.16332671793370550103917829355, −17.76129734715141834183564843820, −16.27497162431184487746693687531, −14.75523843203076855582127601443, −12.84343600036473594049829136511, −11.09807853236429386575962021906, −9.085025602768417102353299621107, −7.57305390382141328359896637786, −4.29114975268875435766358984040, 0,
4.29114975268875435766358984040, 7.57305390382141328359896637786, 9.085025602768417102353299621107, 11.09807853236429386575962021906, 12.84343600036473594049829136511, 14.75523843203076855582127601443, 16.27497162431184487746693687531, 17.76129734715141834183564843820, 19.16332671793370550103917829355
