| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 4·5-s − 6·6-s − 7·7-s + 8·8-s + 9·9-s − 8·10-s − 11·11-s − 12·12-s + 62·13-s − 14·14-s + 12·15-s + 16·16-s − 120·17-s + 18·18-s + 118·19-s − 16·20-s + 21·21-s − 22·22-s − 188·23-s − 24·24-s − 109·25-s + 124·26-s − 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.357·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.252·10-s − 0.301·11-s − 0.288·12-s + 1.32·13-s − 0.267·14-s + 0.206·15-s + 1/4·16-s − 1.71·17-s + 0.235·18-s + 1.42·19-s − 0.178·20-s + 0.218·21-s − 0.213·22-s − 1.70·23-s − 0.204·24-s − 0.871·25-s + 0.935·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 + p T \) |
| 7 | \( 1 + p T \) |
| 11 | \( 1 + p T \) |
| good | 5 | \( 1 + 4 T + p^{3} T^{2} \) |
| 13 | \( 1 - 62 T + p^{3} T^{2} \) |
| 17 | \( 1 + 120 T + p^{3} T^{2} \) |
| 19 | \( 1 - 118 T + p^{3} T^{2} \) |
| 23 | \( 1 + 188 T + p^{3} T^{2} \) |
| 29 | \( 1 - 62 T + p^{3} T^{2} \) |
| 31 | \( 1 + 322 T + p^{3} T^{2} \) |
| 37 | \( 1 + 198 T + p^{3} T^{2} \) |
| 41 | \( 1 - 48 T + p^{3} T^{2} \) |
| 43 | \( 1 - 32 T + p^{3} T^{2} \) |
| 47 | \( 1 + 326 T + p^{3} T^{2} \) |
| 53 | \( 1 + 482 T + p^{3} T^{2} \) |
| 59 | \( 1 - 400 T + p^{3} T^{2} \) |
| 61 | \( 1 - 70 T + p^{3} T^{2} \) |
| 67 | \( 1 + 124 T + p^{3} T^{2} \) |
| 71 | \( 1 + 712 T + p^{3} T^{2} \) |
| 73 | \( 1 - 304 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1016 T + p^{3} T^{2} \) |
| 83 | \( 1 - 430 T + p^{3} T^{2} \) |
| 89 | \( 1 - 442 T + p^{3} T^{2} \) |
| 97 | \( 1 + 966 T + p^{3} T^{2} \) |
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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
−10.47705856939460575426554329063, −9.392570878213424497044094921475, −8.227595559261401325415055267466, −7.18516383089990933970001100669, −6.24897823515503046148939210512, −5.49399535507257054226869669618, −4.26391055053296811740801273198, −3.43176004453482577181423519500, −1.80826015717276569782436375563, 0,
1.80826015717276569782436375563, 3.43176004453482577181423519500, 4.26391055053296811740801273198, 5.49399535507257054226869669618, 6.24897823515503046148939210512, 7.18516383089990933970001100669, 8.227595559261401325415055267466, 9.392570878213424497044094921475, 10.47705856939460575426554329063
