| L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s + 7-s − 8-s + 9-s + 2·10-s + 12-s − 3·13-s − 14-s − 2·15-s + 16-s − 3·17-s − 18-s − 5·19-s − 2·20-s + 21-s + 23-s − 24-s − 25-s + 3·26-s + 27-s + 28-s − 5·29-s + 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.288·12-s − 0.832·13-s − 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 1.14·19-s − 0.447·20-s + 0.218·21-s + 0.208·23-s − 0.204·24-s − 1/5·25-s + 0.588·26-s + 0.192·27-s + 0.188·28-s − 0.928·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.23841657979884, −13.47539158101850, −13.15572080157762, −12.52937287399660, −12.08139866177106, −11.55959288455304, −11.20434825174903, −10.56370404224498, −10.24601735923374, −9.610505899078221, −8.948986872086586, −8.760697650528270, −8.216650840178093, −7.625325655699639, −7.333673325815314, −6.877822443422466, −6.190740172533270, −5.551639185537047, −4.744286320760801, −4.404363570054994, −3.682895926570750, −3.245968103337332, −2.354413485121064, −2.011787431374490, −1.296207782956393, 0, 0,
1.296207782956393, 2.011787431374490, 2.354413485121064, 3.245968103337332, 3.682895926570750, 4.404363570054994, 4.744286320760801, 5.551639185537047, 6.190740172533270, 6.877822443422466, 7.333673325815314, 7.625325655699639, 8.216650840178093, 8.760697650528270, 8.948986872086586, 9.610505899078221, 10.24601735923374, 10.56370404224498, 11.20434825174903, 11.55959288455304, 12.08139866177106, 12.52937287399660, 13.15572080157762, 13.47539158101850, 14.23841657979884
