| L(s) = 1 | − 14·2-s + 68·4-s + 1.64e3·7-s + 840·8-s − 172·11-s − 3.86e3·13-s − 2.30e4·14-s − 2.04e4·16-s − 1.22e4·17-s − 2.59e4·19-s + 2.40e3·22-s + 1.29e4·23-s + 5.40e4·26-s + 1.11e5·28-s + 8.16e4·29-s − 1.56e5·31-s + 1.78e5·32-s + 1.71e5·34-s − 1.10e5·37-s + 3.63e5·38-s − 4.67e5·41-s + 4.99e5·43-s − 1.16e4·44-s − 1.81e5·46-s − 3.96e5·47-s + 1.87e6·49-s − 2.62e5·52-s + ⋯ |
| L(s) = 1 | − 1.23·2-s + 0.531·4-s + 1.81·7-s + 0.580·8-s − 0.0389·11-s − 0.487·13-s − 2.24·14-s − 1.24·16-s − 0.604·17-s − 0.867·19-s + 0.0482·22-s + 0.222·23-s + 0.603·26-s + 0.962·28-s + 0.621·29-s − 0.945·31-s + 0.965·32-s + 0.748·34-s − 0.357·37-s + 1.07·38-s − 1.06·41-s + 0.957·43-s − 0.0206·44-s − 0.275·46-s − 0.557·47-s + 2.28·49-s − 0.259·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 7 p T + p^{7} T^{2} \) |
| 7 | \( 1 - 1644 T + p^{7} T^{2} \) |
| 11 | \( 1 + 172 T + p^{7} T^{2} \) |
| 13 | \( 1 + 3862 T + p^{7} T^{2} \) |
| 17 | \( 1 + 12254 T + p^{7} T^{2} \) |
| 19 | \( 1 + 25940 T + p^{7} T^{2} \) |
| 23 | \( 1 - 564 p T + p^{7} T^{2} \) |
| 29 | \( 1 - 81610 T + p^{7} T^{2} \) |
| 31 | \( 1 + 156888 T + p^{7} T^{2} \) |
| 37 | \( 1 + 110126 T + p^{7} T^{2} \) |
| 41 | \( 1 + 467882 T + p^{7} T^{2} \) |
| 43 | \( 1 - 499208 T + p^{7} T^{2} \) |
| 47 | \( 1 + 396884 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1280498 T + p^{7} T^{2} \) |
| 59 | \( 1 - 1337420 T + p^{7} T^{2} \) |
| 61 | \( 1 + 923978 T + p^{7} T^{2} \) |
| 67 | \( 1 - 797304 T + p^{7} T^{2} \) |
| 71 | \( 1 + 5103392 T + p^{7} T^{2} \) |
| 73 | \( 1 - 4267478 T + p^{7} T^{2} \) |
| 79 | \( 1 + 960 T + p^{7} T^{2} \) |
| 83 | \( 1 - 6140832 T + p^{7} T^{2} \) |
| 89 | \( 1 + 2010570 T + p^{7} T^{2} \) |
| 97 | \( 1 - 4881934 T + p^{7} T^{2} \) |
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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
−10.58651038208196706130852999408, −9.342317723494190576927339941303, −8.473421392889925733511336407841, −7.86865430656889013134500429133, −6.85999430917465431390934259689, −5.15252587340879113562066353529, −4.31893310148058171188744029716, −2.20517991521422499312839618955, −1.34692517842840542426501629975, 0,
1.34692517842840542426501629975, 2.20517991521422499312839618955, 4.31893310148058171188744029716, 5.15252587340879113562066353529, 6.85999430917465431390934259689, 7.86865430656889013134500429133, 8.473421392889925733511336407841, 9.342317723494190576927339941303, 10.58651038208196706130852999408
