| L(s) = 1 | + 3.76·2-s − 7.36·3-s + 6.16·4-s − 15.4·5-s − 27.7·6-s − 7·7-s − 6.90·8-s + 27.2·9-s − 58.3·10-s + 11·11-s − 45.3·12-s + 49.0·13-s − 26.3·14-s + 114.·15-s − 75.3·16-s − 34.1·17-s + 102.·18-s − 144.·19-s − 95.5·20-s + 51.5·21-s + 41.4·22-s + 118.·23-s + 50.8·24-s + 115.·25-s + 184.·26-s − 1.63·27-s − 43.1·28-s + ⋯ |
| L(s) = 1 | + 1.33·2-s − 1.41·3-s + 0.770·4-s − 1.38·5-s − 1.88·6-s − 0.377·7-s − 0.305·8-s + 1.00·9-s − 1.84·10-s + 0.301·11-s − 1.09·12-s + 1.04·13-s − 0.502·14-s + 1.96·15-s − 1.17·16-s − 0.486·17-s + 1.34·18-s − 1.74·19-s − 1.06·20-s + 0.535·21-s + 0.401·22-s + 1.07·23-s + 0.432·24-s + 0.920·25-s + 1.39·26-s − 0.0116·27-s − 0.291·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 + 7T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 3.76T + 8T^{2} \) |
| 3 | \( 1 + 7.36T + 27T^{2} \) |
| 5 | \( 1 + 15.4T + 125T^{2} \) |
| 13 | \( 1 - 49.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 34.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 144.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 118.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 63.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 212.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 200.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 451.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 130.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 176.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 629.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 86.9T + 2.05e5T^{2} \) |
| 61 | \( 1 - 644.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 400.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 507.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 176.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 701.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.25e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 788.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 185.T + 9.12e5T^{2} \) |
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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
−12.98290294102715277677537190096, −12.50885535221936243262932407851, −11.31106202974706013077606134142, −11.00490809111876813323187937030, −8.758606309476125421947229676998, −6.89947703437141300004680603572, −5.98983998150940296546754602579, −4.63810192447921754194094079760, −3.67358803512856225534933034657, 0,
3.67358803512856225534933034657, 4.63810192447921754194094079760, 5.98983998150940296546754602579, 6.89947703437141300004680603572, 8.758606309476125421947229676998, 11.00490809111876813323187937030, 11.31106202974706013077606134142, 12.50885535221936243262932407851, 12.98290294102715277677537190096
