| L(s) = 1 | + 8·2-s + 27·3-s + 64·4-s − 125·5-s + 216·6-s − 677.·7-s + 512·8-s + 729·9-s − 1.00e3·10-s − 1.90e3·11-s + 1.72e3·12-s + 6.01e3·13-s − 5.42e3·14-s − 3.37e3·15-s + 4.09e3·16-s − 8.03e3·17-s + 5.83e3·18-s + 6.85e3·19-s − 8.00e3·20-s − 1.83e4·21-s − 1.52e4·22-s + 3.93e4·23-s + 1.38e4·24-s + 1.56e4·25-s + 4.81e4·26-s + 1.96e4·27-s − 4.33e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s − 0.747·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.432·11-s + 0.288·12-s + 0.759·13-s − 0.528·14-s − 0.258·15-s + 0.250·16-s − 0.396·17-s + 0.235·18-s + 0.229·19-s − 0.223·20-s − 0.431·21-s − 0.305·22-s + 0.674·23-s + 0.204·24-s + 0.199·25-s + 0.536·26-s + 0.192·27-s − 0.373·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 27T \) |
| 5 | \( 1 + 125T \) |
| 19 | \( 1 - 6.85e3T \) |
| good | 7 | \( 1 + 677.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.90e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 6.01e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 8.03e3T + 4.10e8T^{2} \) |
| 23 | \( 1 - 3.93e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 9.19e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.42e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.52e3T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.13e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.80e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.82e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.44e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.35e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.56e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.09e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.75e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.75e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.83e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.56e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.20e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 4.23e6T + 8.07e13T^{2} \) |
| 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
−9.109549827400709251761319278075, −8.237721227463489787891041196509, −7.27797206075667296565897395159, −6.49242486515091343838297449064, −5.43890003251423860442586389228, −4.32055868228538483939500938402, −3.44127182674164568704932858520, −2.72435818668792170036762758670, −1.42015912507782759707320877582, 0,
1.42015912507782759707320877582, 2.72435818668792170036762758670, 3.44127182674164568704932858520, 4.32055868228538483939500938402, 5.43890003251423860442586389228, 6.49242486515091343838297449064, 7.27797206075667296565897395159, 8.237721227463489787891041196509, 9.109549827400709251761319278075
