| L(s) = 1 | − 41.5·2-s − 243·3-s − 318.·4-s + 9.65e3·5-s + 1.01e4·6-s − 1.15e4·7-s + 9.84e4·8-s + 5.90e4·9-s − 4.01e5·10-s − 5.21e5·11-s + 7.74e4·12-s + 2.19e6·13-s + 4.81e5·14-s − 2.34e6·15-s − 3.44e6·16-s − 5.78e6·17-s − 2.45e6·18-s + 1.37e7·19-s − 3.07e6·20-s + 2.81e6·21-s + 2.16e7·22-s + 5.21e7·23-s − 2.39e7·24-s + 4.44e7·25-s − 9.11e7·26-s − 1.43e7·27-s + 3.68e6·28-s + ⋯ |
| L(s) = 1 | − 0.918·2-s − 0.577·3-s − 0.155·4-s + 1.38·5-s + 0.530·6-s − 0.260·7-s + 1.06·8-s + 0.333·9-s − 1.27·10-s − 0.975·11-s + 0.0898·12-s + 1.63·13-s + 0.239·14-s − 0.797·15-s − 0.820·16-s − 0.987·17-s − 0.306·18-s + 1.27·19-s − 0.214·20-s + 0.150·21-s + 0.896·22-s + 1.68·23-s − 0.613·24-s + 0.910·25-s − 1.50·26-s − 0.192·27-s + 0.0405·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.448316240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.448316240\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 243T \) |
| 59 | \( 1 + 7.14e8T \) |
| good | 2 | \( 1 + 41.5T + 2.04e3T^{2} \) |
| 5 | \( 1 - 9.65e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 1.15e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 5.21e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 2.19e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 5.78e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.37e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 5.21e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.77e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.22e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 3.62e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.34e9T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.17e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 5.28e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 5.18e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 9.00e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 5.11e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.99e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 2.74e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 2.06e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 3.14e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 2.32e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.39e11T + 7.15e21T^{2} \) |
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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
−10.67802745371422872550632550573, −9.508286507532419192306921186383, −9.023001390762507990333990125114, −7.75986319963808155169799072030, −6.51388412290240077151062621600, −5.60750254134643146281308154156, −4.59609532717405677905626759074, −2.83355099984316624429489561620, −1.43972999000294684773125356216, −0.72778073072673147384900847899,
0.72778073072673147384900847899, 1.43972999000294684773125356216, 2.83355099984316624429489561620, 4.59609532717405677905626759074, 5.60750254134643146281308154156, 6.51388412290240077151062621600, 7.75986319963808155169799072030, 9.023001390762507990333990125114, 9.508286507532419192306921186383, 10.67802745371422872550632550573
