| L(s) = 1 | + 8·2-s − 27·3-s + 64·4-s + 200.·5-s − 216·6-s − 286·7-s + 512·8-s + 729·9-s + 1.60e3·10-s − 6.73e3·11-s − 1.72e3·12-s − 161.·13-s − 2.28e3·14-s − 5.40e3·15-s + 4.09e3·16-s − 1.90e4·17-s + 5.83e3·18-s + 4.03e4·19-s + 1.28e4·20-s + 7.72e3·21-s − 5.38e4·22-s − 1.21e4·23-s − 1.38e4·24-s − 3.80e4·25-s − 1.29e3·26-s − 1.96e4·27-s − 1.83e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.716·5-s − 0.408·6-s − 0.315·7-s + 0.353·8-s + 0.333·9-s + 0.506·10-s − 1.52·11-s − 0.288·12-s − 0.0203·13-s − 0.222·14-s − 0.413·15-s + 0.250·16-s − 0.939·17-s + 0.235·18-s + 1.35·19-s + 0.358·20-s + 0.181·21-s − 1.07·22-s − 0.208·23-s − 0.204·24-s − 0.486·25-s − 0.0144·26-s − 0.192·27-s − 0.157·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 27T \) |
| 23 | \( 1 + 1.21e4T \) |
| good | 5 | \( 1 - 200.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 286T + 8.23e5T^{2} \) |
| 11 | \( 1 + 6.73e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 161.T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.90e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.03e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 1.58e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 9.05e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.09e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.26e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.14e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.90e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.02e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 8.07e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.07e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.22e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.18e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.93e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.27e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 9.51e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.66e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 5.15e6T + 8.07e13T^{2} \) |
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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
−11.39338483710664844897935178925, −10.44662553797160600482080230864, −9.553549056069846306668223818657, −7.889431582517759639854041030447, −6.69508249662584098501819474713, −5.62564690441765406836878312744, −4.85849258587799177294765055317, −3.17127092814312610250792824418, −1.85260559868264223748567473408, 0,
1.85260559868264223748567473408, 3.17127092814312610250792824418, 4.85849258587799177294765055317, 5.62564690441765406836878312744, 6.69508249662584098501819474713, 7.889431582517759639854041030447, 9.553549056069846306668223818657, 10.44662553797160600482080230864, 11.39338483710664844897935178925
