| L(s) = 1 | − 8·2-s + 63·3-s + 64·4-s − 504·6-s − 343·7-s − 512·8-s + 1.78e3·9-s − 2.72e3·11-s + 4.03e3·12-s − 5.26e3·13-s + 2.74e3·14-s + 4.09e3·16-s + 1.77e4·17-s − 1.42e4·18-s + 712·19-s − 2.16e4·21-s + 2.18e4·22-s + 2.93e4·23-s − 3.22e4·24-s + 4.21e4·26-s − 2.55e4·27-s − 2.19e4·28-s + 6.84e4·29-s + 1.85e5·31-s − 3.27e4·32-s − 1.71e5·33-s − 1.41e5·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.34·3-s + 1/2·4-s − 0.952·6-s − 0.377·7-s − 0.353·8-s + 0.814·9-s − 0.617·11-s + 0.673·12-s − 0.665·13-s + 0.267·14-s + 1/4·16-s + 0.873·17-s − 0.576·18-s + 0.0238·19-s − 0.509·21-s + 0.436·22-s + 0.502·23-s − 0.476·24-s + 0.470·26-s − 0.249·27-s − 0.188·28-s + 0.521·29-s + 1.11·31-s − 0.176·32-s − 0.832·33-s − 0.617·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p^{3} T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p^{3} T \) |
| good | 3 | \( 1 - 7 p^{2} T + p^{7} T^{2} \) |
| 11 | \( 1 + 2727 T + p^{7} T^{2} \) |
| 13 | \( 1 + 5269 T + p^{7} T^{2} \) |
| 17 | \( 1 - 17701 T + p^{7} T^{2} \) |
| 19 | \( 1 - 712 T + p^{7} T^{2} \) |
| 23 | \( 1 - 29330 T + p^{7} T^{2} \) |
| 29 | \( 1 - 68491 T + p^{7} T^{2} \) |
| 31 | \( 1 - 185026 T + p^{7} T^{2} \) |
| 37 | \( 1 - 6758 p T + p^{7} T^{2} \) |
| 41 | \( 1 + 125814 T + p^{7} T^{2} \) |
| 43 | \( 1 + 747476 T + p^{7} T^{2} \) |
| 47 | \( 1 + 317317 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1623246 T + p^{7} T^{2} \) |
| 59 | \( 1 + 1519262 T + p^{7} T^{2} \) |
| 61 | \( 1 + 54240 p T + p^{7} T^{2} \) |
| 67 | \( 1 - 2272366 T + p^{7} T^{2} \) |
| 71 | \( 1 + 4963104 T + p^{7} T^{2} \) |
| 73 | \( 1 + 2351750 T + p^{7} T^{2} \) |
| 79 | \( 1 + 2524249 T + p^{7} T^{2} \) |
| 83 | \( 1 + 6051492 T + p^{7} T^{2} \) |
| 89 | \( 1 - 8043880 T + p^{7} T^{2} \) |
| 97 | \( 1 - 2337645 T + p^{7} T^{2} \) |
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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
−9.744277142406022481958025357950, −8.869945153407033430357105451697, −8.020730222753910653890846736441, −7.43267196587194912865059910446, −6.22090412586055502684768309776, −4.78141411401163709592320028132, −3.23223147212647533760477151269, −2.69676092768815901449607449781, −1.45056700901309175751873594479, 0,
1.45056700901309175751873594479, 2.69676092768815901449607449781, 3.23223147212647533760477151269, 4.78141411401163709592320028132, 6.22090412586055502684768309776, 7.43267196587194912865059910446, 8.020730222753910653890846736441, 8.869945153407033430357105451697, 9.744277142406022481958025357950
