| L(s) = 1 | − 1.62·2-s − 9·3-s − 29.3·4-s + 103.·5-s + 14.6·6-s − 137.·7-s + 99.6·8-s + 81·9-s − 168.·10-s − 636.·11-s + 264.·12-s + 834.·13-s + 223.·14-s − 931.·15-s + 777.·16-s + 444.·17-s − 131.·18-s − 513.·19-s − 3.03e3·20-s + 1.23e3·21-s + 1.03e3·22-s + 3.56e3·23-s − 896.·24-s + 7.58e3·25-s − 1.35e3·26-s − 729·27-s + 4.03e3·28-s + ⋯ |
| L(s) = 1 | − 0.286·2-s − 0.577·3-s − 0.917·4-s + 1.85·5-s + 0.165·6-s − 1.06·7-s + 0.550·8-s + 0.333·9-s − 0.531·10-s − 1.58·11-s + 0.529·12-s + 1.36·13-s + 0.304·14-s − 1.06·15-s + 0.759·16-s + 0.373·17-s − 0.0956·18-s − 0.326·19-s − 1.69·20-s + 0.612·21-s + 0.455·22-s + 1.40·23-s − 0.317·24-s + 2.42·25-s − 0.393·26-s − 0.192·27-s + 0.973·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 9T \) |
| 59 | \( 1 + 3.48e3T \) |
| good | 2 | \( 1 + 1.62T + 32T^{2} \) |
| 5 | \( 1 - 103.T + 3.12e3T^{2} \) |
| 7 | \( 1 + 137.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 636.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 834.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 444.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 513.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.56e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.87e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.63e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.91e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.99e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.56e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.22e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.99e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 3.45e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.93e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.24e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.83e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.92e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.04e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.01e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.16e4T + 8.58e9T^{2} \) |
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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
−10.82448371106372691027132684861, −10.21122225114023951663554175452, −9.457210209782953348957533441084, −8.556363064368876338312328639147, −6.88016311660453653505743444450, −5.71187175624424826809195908778, −5.19819714254169550445900259959, −3.25979847504843219389997226897, −1.53298621313985742549708516419, 0,
1.53298621313985742549708516419, 3.25979847504843219389997226897, 5.19819714254169550445900259959, 5.71187175624424826809195908778, 6.88016311660453653505743444450, 8.556363064368876338312328639147, 9.457210209782953348957533441084, 10.21122225114023951663554175452, 10.82448371106372691027132684861
