| L(s) = 1 | + 8·2-s + 5.57·3-s + 64·4-s + 202.·5-s + 44.5·6-s + 343·7-s + 512·8-s − 2.15e3·9-s + 1.61e3·10-s − 7.17e3·11-s + 356.·12-s + 2.19e3·13-s + 2.74e3·14-s + 1.12e3·15-s + 4.09e3·16-s − 2.04e4·17-s − 1.72e4·18-s − 2.49e4·19-s + 1.29e4·20-s + 1.91e3·21-s − 5.73e4·22-s − 4.75e4·23-s + 2.85e3·24-s − 3.72e4·25-s + 1.75e4·26-s − 2.41e4·27-s + 2.19e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.119·3-s + 0.5·4-s + 0.723·5-s + 0.0842·6-s + 0.377·7-s + 0.353·8-s − 0.985·9-s + 0.511·10-s − 1.62·11-s + 0.0595·12-s + 0.277·13-s + 0.267·14-s + 0.0861·15-s + 0.250·16-s − 1.01·17-s − 0.697·18-s − 0.833·19-s + 0.361·20-s + 0.0450·21-s − 1.14·22-s − 0.814·23-s + 0.0421·24-s − 0.476·25-s + 0.196·26-s − 0.236·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 8T \) |
| 7 | \( 1 - 343T \) |
| 13 | \( 1 - 2.19e3T \) |
| good | 3 | \( 1 - 5.57T + 2.18e3T^{2} \) |
| 5 | \( 1 - 202.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 7.17e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 2.04e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.49e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 4.75e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.69e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.40e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.18e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.69e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 9.74e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 8.34e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 9.07e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.85e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 7.65e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.10e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.73e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 6.25e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.55e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 9.36e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.87e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.31e7T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.92189782728015444526392846040, −10.09182665156218400292802471161, −8.663499744614423501793244734839, −7.80205125615303642213397496759, −6.28697599321958993434131808798, −5.53419389157889640175379223909, −4.42956694516801084552287865809, −2.80989787631468106109488854796, −2.01621739991831994374551913532, 0,
2.01621739991831994374551913532, 2.80989787631468106109488854796, 4.42956694516801084552287865809, 5.53419389157889640175379223909, 6.28697599321958993434131808798, 7.80205125615303642213397496759, 8.663499744614423501793244734839, 10.09182665156218400292802471161, 10.92189782728015444526392846040
