| L(s) = 1 | − 4.14·2-s − 9·3-s − 14.8·4-s − 34.9·5-s + 37.3·6-s − 110.·7-s + 194.·8-s + 81·9-s + 144.·10-s + 176.·11-s + 133.·12-s + 818.·13-s + 456.·14-s + 314.·15-s − 330.·16-s − 1.40e3·17-s − 335.·18-s + 2.00e3·19-s + 518.·20-s + 992.·21-s − 731.·22-s + 846.·23-s − 1.74e3·24-s − 1.90e3·25-s − 3.39e3·26-s − 729·27-s + 1.63e3·28-s + ⋯ |
| L(s) = 1 | − 0.732·2-s − 0.577·3-s − 0.463·4-s − 0.625·5-s + 0.423·6-s − 0.850·7-s + 1.07·8-s + 0.333·9-s + 0.458·10-s + 0.439·11-s + 0.267·12-s + 1.34·13-s + 0.623·14-s + 0.361·15-s − 0.322·16-s − 1.17·17-s − 0.244·18-s + 1.27·19-s + 0.289·20-s + 0.490·21-s − 0.322·22-s + 0.333·23-s − 0.618·24-s − 0.608·25-s − 0.984·26-s − 0.192·27-s + 0.393·28-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 | \( 1 + 4.14T + 32T^{2} \) |
| 5 | \( 1 + 34.9T + 3.12e3T^{2} \) |
| 7 | \( 1 + 110.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 176.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 818.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.40e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.00e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 846.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.64e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.74e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.23e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.19e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.37e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 7.63e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.03e3T + 4.18e8T^{2} \) |
| 61 | \( 1 + 4.02e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.42e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.52e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.86e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.50e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.23e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.36e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.75e4T + 8.58e9T^{2} \) |
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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
−11.18571750431717269063635706572, −10.23834045895186853798515153230, −9.243790533994873278716774799161, −8.410452142799547128868213382491, −7.18671892250258769450531154681, −6.13207414298376202375727478417, −4.62079440547774134004391208679, −3.51281369855920537011636280744, −1.17231383859425890714673088570, 0,
1.17231383859425890714673088570, 3.51281369855920537011636280744, 4.62079440547774134004391208679, 6.13207414298376202375727478417, 7.18671892250258769450531154681, 8.410452142799547128868213382491, 9.243790533994873278716774799161, 10.23834045895186853798515153230, 11.18571750431717269063635706572
