| L(s) = 1 | + 8·2-s − 39·3-s + 64·4-s − 385·5-s − 312·6-s + 293·7-s + 512·8-s − 666·9-s − 3.08e3·10-s + 5.40e3·11-s − 2.49e3·12-s + 2.34e3·14-s + 1.50e4·15-s + 4.09e3·16-s − 2.10e4·17-s − 5.32e3·18-s + 2.73e4·19-s − 2.46e4·20-s − 1.14e4·21-s + 4.32e4·22-s − 6.30e4·23-s − 1.99e4·24-s + 7.01e4·25-s + 1.11e5·27-s + 1.87e4·28-s + 1.22e5·29-s + 1.20e5·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.833·3-s + 1/2·4-s − 1.37·5-s − 0.589·6-s + 0.322·7-s + 0.353·8-s − 0.304·9-s − 0.973·10-s + 1.22·11-s − 0.416·12-s + 0.228·14-s + 1.14·15-s + 1/4·16-s − 1.03·17-s − 0.215·18-s + 0.913·19-s − 0.688·20-s − 0.269·21-s + 0.865·22-s − 1.08·23-s − 0.294·24-s + 0.897·25-s + 1.08·27-s + 0.161·28-s + 0.930·29-s + 0.812·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{3} T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 13 p T + p^{7} T^{2} \) |
| 5 | \( 1 + 77 p T + p^{7} T^{2} \) |
| 7 | \( 1 - 293 T + p^{7} T^{2} \) |
| 11 | \( 1 - 5402 T + p^{7} T^{2} \) |
| 17 | \( 1 + 21011 T + p^{7} T^{2} \) |
| 19 | \( 1 - 27326 T + p^{7} T^{2} \) |
| 23 | \( 1 + 63072 T + p^{7} T^{2} \) |
| 29 | \( 1 - 122238 T + p^{7} T^{2} \) |
| 31 | \( 1 - 208396 T + p^{7} T^{2} \) |
| 37 | \( 1 - 442379 T + p^{7} T^{2} \) |
| 41 | \( 1 + 58000 T + p^{7} T^{2} \) |
| 43 | \( 1 + 202025 T + p^{7} T^{2} \) |
| 47 | \( 1 + 588511 T + p^{7} T^{2} \) |
| 53 | \( 1 - 1684336 T + p^{7} T^{2} \) |
| 59 | \( 1 - 442630 T + p^{7} T^{2} \) |
| 61 | \( 1 + 1083608 T + p^{7} T^{2} \) |
| 67 | \( 1 + 3443486 T + p^{7} T^{2} \) |
| 71 | \( 1 + 2084705 T + p^{7} T^{2} \) |
| 73 | \( 1 + 5937890 T + p^{7} T^{2} \) |
| 79 | \( 1 + 6609256 T + p^{7} T^{2} \) |
| 83 | \( 1 - 142740 T + p^{7} T^{2} \) |
| 89 | \( 1 - 6985286 T + p^{7} T^{2} \) |
| 97 | \( 1 - 200762 T + p^{7} T^{2} \) |
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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
−10.19924886708937658345058054494, −8.749459003041234493272574292757, −7.84558802713641637664207295412, −6.75944354149565569816490411719, −6.00277912395252041070884980260, −4.68944135974671314254619063678, −4.11899500655301060813880163470, −2.90571910132255091115667925305, −1.18157401308430318190862061395, 0,
1.18157401308430318190862061395, 2.90571910132255091115667925305, 4.11899500655301060813880163470, 4.68944135974671314254619063678, 6.00277912395252041070884980260, 6.75944354149565569816490411719, 7.84558802713641637664207295412, 8.749459003041234493272574292757, 10.19924886708937658345058054494
