| L(s) = 1 | − 12.7·2-s + 81·3-s − 349.·4-s + 1.73e3·5-s − 1.03e3·6-s + 7.50e3·7-s + 1.09e4·8-s + 6.56e3·9-s − 2.21e4·10-s − 7.38e4·11-s − 2.82e4·12-s + 1.62e5·13-s − 9.58e4·14-s + 1.40e5·15-s + 3.86e4·16-s − 6.63e5·17-s − 8.37e4·18-s − 7.16e5·19-s − 6.07e5·20-s + 6.08e5·21-s + 9.42e5·22-s − 1.59e6·23-s + 8.89e5·24-s + 1.07e6·25-s − 2.07e6·26-s + 5.31e5·27-s − 2.62e6·28-s + ⋯ |
| L(s) = 1 | − 0.563·2-s + 0.577·3-s − 0.682·4-s + 1.24·5-s − 0.325·6-s + 1.18·7-s + 0.948·8-s + 0.333·9-s − 0.701·10-s − 1.52·11-s − 0.393·12-s + 1.57·13-s − 0.666·14-s + 0.718·15-s + 0.147·16-s − 1.92·17-s − 0.187·18-s − 1.26·19-s − 0.848·20-s + 0.682·21-s + 0.857·22-s − 1.18·23-s + 0.547·24-s + 0.548·25-s − 0.890·26-s + 0.192·27-s − 0.806·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 59 | \( 1 - 1.21e7T \) |
| good | 2 | \( 1 + 12.7T + 512T^{2} \) |
| 5 | \( 1 - 1.73e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 7.50e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 7.38e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.62e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 6.63e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 7.16e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.59e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.50e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.91e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.43e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.18e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.63e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.19e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.54e7T + 3.29e15T^{2} \) |
| 61 | \( 1 + 1.00e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.65e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.72e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 5.63e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.55e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.01e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 4.74e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 5.39e8T + 7.60e17T^{2} \) |
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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
−10.53019620551108621707378742806, −9.325243704008694178050359277305, −8.484042447899382027633922832627, −7.984734465513379970886716512002, −6.31012261964002294271726405860, −5.11028355413849834899304693562, −4.12559708812292692829660607265, −2.22073629081034600062172336681, −1.61229660213092713184074512018, 0,
1.61229660213092713184074512018, 2.22073629081034600062172336681, 4.12559708812292692829660607265, 5.11028355413849834899304693562, 6.31012261964002294271726405860, 7.984734465513379970886716512002, 8.484042447899382027633922832627, 9.325243704008694178050359277305, 10.53019620551108621707378742806
