| L(s) = 1 | + 3.55·2-s + 0.389·3-s + 4.66·4-s + 1.72·5-s + 1.38·6-s + 21.1·7-s − 11.8·8-s − 26.8·9-s + 6.14·10-s + 23.7·11-s + 1.81·12-s − 30.1·13-s + 75.3·14-s + 0.672·15-s − 79.5·16-s + 51.6·17-s − 95.5·18-s − 29.2·19-s + 8.05·20-s + 8.24·21-s + 84.5·22-s − 93.4·23-s − 4.62·24-s − 122.·25-s − 107.·26-s − 20.9·27-s + 98.7·28-s + ⋯ |
| L(s) = 1 | + 1.25·2-s + 0.0748·3-s + 0.582·4-s + 0.154·5-s + 0.0942·6-s + 1.14·7-s − 0.524·8-s − 0.994·9-s + 0.194·10-s + 0.650·11-s + 0.0436·12-s − 0.643·13-s + 1.43·14-s + 0.0115·15-s − 1.24·16-s + 0.737·17-s − 1.25·18-s − 0.353·19-s + 0.0900·20-s + 0.0856·21-s + 0.818·22-s − 0.847·23-s − 0.0393·24-s − 0.976·25-s − 0.810·26-s − 0.149·27-s + 0.666·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - 3.55T + 8T^{2} \) |
| 3 | \( 1 - 0.389T + 27T^{2} \) |
| 5 | \( 1 - 1.72T + 125T^{2} \) |
| 7 | \( 1 - 21.1T + 343T^{2} \) |
| 11 | \( 1 - 23.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 30.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 51.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 29.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 93.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 59.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 113.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 68.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 53.9T + 6.89e4T^{2} \) |
| 47 | \( 1 + 455.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 662.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 457.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 606.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 428.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 139.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 481.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.10e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.25e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 624.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.12e3T + 9.12e5T^{2} \) |
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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
−8.330617237438646111562044502689, −7.79631280528819195028651573123, −6.54044013908199098433348796574, −5.89068309933503658792025425688, −5.12356791268222102430033843713, −4.48209194138413447134103684881, −3.57418321512766413909394431252, −2.63800861356743480732694599697, −1.63727578416651994153871310576, 0,
1.63727578416651994153871310576, 2.63800861356743480732694599697, 3.57418321512766413909394431252, 4.48209194138413447134103684881, 5.12356791268222102430033843713, 5.89068309933503658792025425688, 6.54044013908199098433348796574, 7.79631280528819195028651573123, 8.330617237438646111562044502689
