| L(s) = 1 | − 8·2-s − 27·3-s + 64·4-s + 458.·5-s + 216·6-s − 283.·7-s − 512·8-s + 729·9-s − 3.66e3·10-s − 2.81e3·11-s − 1.72e3·12-s − 467.·13-s + 2.27e3·14-s − 1.23e4·15-s + 4.09e3·16-s − 6.78e3·17-s − 5.83e3·18-s − 5.10e4·19-s + 2.93e4·20-s + 7.66e3·21-s + 2.25e4·22-s + 1.21e4·23-s + 1.38e4·24-s + 1.32e5·25-s + 3.73e3·26-s − 1.96e4·27-s − 1.81e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.64·5-s + 0.408·6-s − 0.312·7-s − 0.353·8-s + 0.333·9-s − 1.16·10-s − 0.638·11-s − 0.288·12-s − 0.0589·13-s + 0.221·14-s − 0.947·15-s + 0.250·16-s − 0.334·17-s − 0.235·18-s − 1.70·19-s + 0.820·20-s + 0.180·21-s + 0.451·22-s + 0.208·23-s + 0.204·24-s + 1.69·25-s + 0.0417·26-s − 0.192·27-s − 0.156·28-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 | \( 1 - 458.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 283.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 2.81e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 467.T + 6.27e7T^{2} \) |
| 17 | \( 1 + 6.78e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.10e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 2.17e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 3.07e3T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.51e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.24e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.20e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 6.02e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.93e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.19e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.24e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.54e4T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.73e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.17e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.12e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.08e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 7.45e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.22e6T + 8.07e13T^{2} \) |
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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
−10.89039273621200203408261336117, −10.29856396541775290926016550349, −9.440461200216178035016359822696, −8.356176297516385650796882165630, −6.71135832081796534180794397414, −6.10193439101673276576662346290, −4.86899363091464727003716435773, −2.65032817347850084027935859613, −1.55465108699675397321536164138, 0,
1.55465108699675397321536164138, 2.65032817347850084027935859613, 4.86899363091464727003716435773, 6.10193439101673276576662346290, 6.71135832081796534180794397414, 8.356176297516385650796882165630, 9.440461200216178035016359822696, 10.29856396541775290926016550349, 10.89039273621200203408261336117
