| L(s) = 1 | + 13.0·2-s − 27·3-s + 41.1·4-s + 167.·5-s − 351.·6-s + 887.·7-s − 1.13e3·8-s + 729·9-s + 2.17e3·10-s − 7.78e3·11-s − 1.10e3·12-s + 4.60e3·13-s + 1.15e4·14-s − 4.51e3·15-s − 1.99e4·16-s − 6.05e3·17-s + 9.47e3·18-s + 1.35e4·19-s + 6.87e3·20-s − 2.39e4·21-s − 1.01e5·22-s + 3.52e4·23-s + 3.05e4·24-s − 5.01e4·25-s + 5.99e4·26-s − 1.96e4·27-s + 3.64e4·28-s + ⋯ |
| L(s) = 1 | + 1.14·2-s − 0.577·3-s + 0.321·4-s + 0.598·5-s − 0.663·6-s + 0.977·7-s − 0.780·8-s + 0.333·9-s + 0.687·10-s − 1.76·11-s − 0.185·12-s + 0.581·13-s + 1.12·14-s − 0.345·15-s − 1.21·16-s − 0.298·17-s + 0.383·18-s + 0.453·19-s + 0.192·20-s − 0.564·21-s − 2.02·22-s + 0.604·23-s + 0.450·24-s − 0.641·25-s + 0.668·26-s − 0.192·27-s + 0.313·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 59 | \( 1 - 2.05e5T \) |
| good | 2 | \( 1 - 13.0T + 128T^{2} \) |
| 5 | \( 1 - 167.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 887.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 7.78e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 4.60e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 6.05e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.35e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.52e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 7.83e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.70e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.63e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.98e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.44e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.31e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.00e6T + 1.17e12T^{2} \) |
| 61 | \( 1 + 1.77e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.37e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.87e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.90e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.66e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.18e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.13e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 2.43e5T + 8.07e13T^{2} \) |
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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
−11.09982804502068330048081301107, −10.21490348764555449053828058791, −8.828305460416954273068638644758, −7.61880872109579581776510525867, −6.18364832931992730942427805891, −5.26827615001935151750828759923, −4.74652432992454817465707469429, −3.18196013146654100597708254876, −1.79272410444389339094485643461, 0,
1.79272410444389339094485643461, 3.18196013146654100597708254876, 4.74652432992454817465707469429, 5.26827615001935151750828759923, 6.18364832931992730942427805891, 7.61880872109579581776510525867, 8.828305460416954273068638644758, 10.21490348764555449053828058791, 11.09982804502068330048081301107
