| L(s) = 1 | + 2·2-s − 7·3-s + 4·4-s − 15·5-s − 14·6-s − 18·7-s + 8·8-s + 22·9-s − 30·10-s + 27·11-s − 28·12-s − 57·13-s − 36·14-s + 105·15-s + 16·16-s − 44·17-s + 44·18-s + 152·19-s − 60·20-s + 126·21-s + 54·22-s − 152·23-s − 56·24-s + 100·25-s − 114·26-s + 35·27-s − 72·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.34·3-s + 1/2·4-s − 1.34·5-s − 0.952·6-s − 0.971·7-s + 0.353·8-s + 0.814·9-s − 0.948·10-s + 0.740·11-s − 0.673·12-s − 1.21·13-s − 0.687·14-s + 1.80·15-s + 1/4·16-s − 0.627·17-s + 0.576·18-s + 1.83·19-s − 0.670·20-s + 1.30·21-s + 0.523·22-s − 1.37·23-s − 0.476·24-s + 4/5·25-s − 0.859·26-s + 0.249·27-s − 0.485·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 29 | \( 1 + p T \) |
| good | 3 | \( 1 + 7 T + p^{3} T^{2} \) |
| 5 | \( 1 + 3 p T + p^{3} T^{2} \) |
| 7 | \( 1 + 18 T + p^{3} T^{2} \) |
| 11 | \( 1 - 27 T + p^{3} T^{2} \) |
| 13 | \( 1 + 57 T + p^{3} T^{2} \) |
| 17 | \( 1 + 44 T + p^{3} T^{2} \) |
| 19 | \( 1 - 8 p T + p^{3} T^{2} \) |
| 23 | \( 1 + 152 T + p^{3} T^{2} \) |
| 31 | \( 1 + 173 T + p^{3} T^{2} \) |
| 37 | \( 1 + 120 T + p^{3} T^{2} \) |
| 41 | \( 1 + 314 T + p^{3} T^{2} \) |
| 43 | \( 1 - 339 T + p^{3} T^{2} \) |
| 47 | \( 1 + 357 T + p^{3} T^{2} \) |
| 53 | \( 1 + 59 T + p^{3} T^{2} \) |
| 59 | \( 1 + 572 T + p^{3} T^{2} \) |
| 61 | \( 1 + 420 T + p^{3} T^{2} \) |
| 67 | \( 1 - 660 T + p^{3} T^{2} \) |
| 71 | \( 1 - 726 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1004 T + p^{3} T^{2} \) |
| 79 | \( 1 - 361 T + p^{3} T^{2} \) |
| 83 | \( 1 + 168 T + p^{3} T^{2} \) |
| 89 | \( 1 - 58 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1206 T + p^{3} T^{2} \) |
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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
−14.10770035291452106959417370614, −12.47593466139884812603961352545, −11.97755658380684122247894273886, −11.17654726324899574401892344988, −9.703777889700298688354666064558, −7.51348112891113960106021454255, −6.46709742048210118727660070360, −5.05182173109708881828653317664, −3.63166378343343922901328708161, 0,
3.63166378343343922901328708161, 5.05182173109708881828653317664, 6.46709742048210118727660070360, 7.51348112891113960106021454255, 9.703777889700298688354666064558, 11.17654726324899574401892344988, 11.97755658380684122247894273886, 12.47593466139884812603961352545, 14.10770035291452106959417370614
