L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s − 12.1·5-s − 216·6-s + 343·7-s − 512·8-s + 729·9-s + 97.2·10-s + 2.35e3·11-s + 1.72e3·12-s + 2.19e3·13-s − 2.74e3·14-s − 328.·15-s + 4.09e3·16-s − 2.37e4·17-s − 5.83e3·18-s + 4.09e4·19-s − 777.·20-s + 9.26e3·21-s − 1.88e4·22-s − 3.67e4·23-s − 1.38e4·24-s − 7.79e4·25-s − 1.75e4·26-s + 1.96e4·27-s + 2.19e4·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.0434·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.0307·10-s + 0.532·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.0250·15-s + 0.250·16-s − 1.17·17-s − 0.235·18-s + 1.36·19-s − 0.0217·20-s + 0.218·21-s − 0.376·22-s − 0.629·23-s − 0.204·24-s − 0.998·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 8T \) |
| 3 | \( 1 - 27T \) |
| 7 | \( 1 - 343T \) |
| 13 | \( 1 - 2.19e3T \) |
good | 5 | \( 1 + 12.1T + 7.81e4T^{2} \) |
| 11 | \( 1 - 2.35e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 2.37e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.09e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.67e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.45e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 9.10e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.89e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.92e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.08e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.19e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.59e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 6.97e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.63e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.94e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.79e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.60e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.23e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.05e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.20e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.71e7T + 8.07e13T^{2} \) |
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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
−9.436418970929010778693519673568, −8.283131754249614096352439242225, −7.69949947383785125403961690196, −6.73596319756174655550920599408, −5.70790499545174414802055932610, −4.36611303493675314780663792460, −3.36878125568649746244023792179, −2.14776007743985102504299177008, −1.32114274971617126046260370601, 0,
1.32114274971617126046260370601, 2.14776007743985102504299177008, 3.36878125568649746244023792179, 4.36611303493675314780663792460, 5.70790499545174414802055932610, 6.73596319756174655550920599408, 7.69949947383785125403961690196, 8.283131754249614096352439242225, 9.436418970929010778693519673568