| L(s) = 1 | − 4·2-s − 19.3·3-s + 16·4-s + 77.2·6-s − 49·7-s − 64·8-s + 129.·9-s − 10.9·11-s − 308.·12-s − 29.6·13-s + 196·14-s + 256·16-s + 432.·17-s − 518.·18-s − 956.·19-s + 945.·21-s + 43.6·22-s − 979.·23-s + 1.23e3·24-s + 118.·26-s + 2.19e3·27-s − 784·28-s + 996.·29-s + 4.79e3·31-s − 1.02e3·32-s + 210.·33-s − 1.73e3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.23·3-s + 0.5·4-s + 0.875·6-s − 0.377·7-s − 0.353·8-s + 0.532·9-s − 0.0271·11-s − 0.619·12-s − 0.0487·13-s + 0.267·14-s + 0.250·16-s + 0.362·17-s − 0.376·18-s − 0.607·19-s + 0.467·21-s + 0.0192·22-s − 0.386·23-s + 0.437·24-s + 0.0344·26-s + 0.578·27-s − 0.188·28-s + 0.220·29-s + 0.895·31-s − 0.176·32-s + 0.0336·33-s − 0.256·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 + 19.3T + 243T^{2} \) |
| 11 | \( 1 + 10.9T + 1.61e5T^{2} \) |
| 13 | \( 1 + 29.6T + 3.71e5T^{2} \) |
| 17 | \( 1 - 432.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 956.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 979.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 996.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.79e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.88e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.92e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.80e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.85e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 287.T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.12e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.28e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.70e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.39e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.91e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.12e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.43e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.86e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.34e4T + 8.58e9T^{2} \) |
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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
−10.38868304814197244864684583032, −9.411075551449307212161340356311, −8.355236588551091043266516771624, −7.23563766348301664090699992769, −6.28773082868192240715837114685, −5.58787527741936466887580722342, −4.26467838631811684402604213290, −2.64812041274584959464628672297, −1.04486655230655723200970365918, 0,
1.04486655230655723200970365918, 2.64812041274584959464628672297, 4.26467838631811684402604213290, 5.58787527741936466887580722342, 6.28773082868192240715837114685, 7.23563766348301664090699992769, 8.355236588551091043266516771624, 9.411075551449307212161340356311, 10.38868304814197244864684583032
