| L(s) = 1 | − 2.89·2-s + 6.20·3-s + 0.399·4-s − 21.5·5-s − 17.9·6-s − 25.5·7-s + 22.0·8-s + 11.5·9-s + 62.3·10-s − 44.3·11-s + 2.48·12-s + 22.5·13-s + 74.1·14-s − 133.·15-s − 67.0·16-s − 32.0·17-s − 33.3·18-s + 76.6·19-s − 8.60·20-s − 158.·21-s + 128.·22-s + 50.4·23-s + 136.·24-s + 337.·25-s − 65.4·26-s − 96.1·27-s − 10.2·28-s + ⋯ |
| L(s) = 1 | − 1.02·2-s + 1.19·3-s + 0.0499·4-s − 1.92·5-s − 1.22·6-s − 1.38·7-s + 0.973·8-s + 0.426·9-s + 1.97·10-s − 1.21·11-s + 0.0597·12-s + 0.481·13-s + 1.41·14-s − 2.29·15-s − 1.04·16-s − 0.456·17-s − 0.436·18-s + 0.925·19-s − 0.0961·20-s − 1.65·21-s + 1.24·22-s + 0.456·23-s + 1.16·24-s + 2.69·25-s − 0.493·26-s − 0.685·27-s − 0.0690·28-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 | \( 1 + 2.89T + 8T^{2} \) |
| 3 | \( 1 - 6.20T + 27T^{2} \) |
| 5 | \( 1 + 21.5T + 125T^{2} \) |
| 7 | \( 1 + 25.5T + 343T^{2} \) |
| 11 | \( 1 + 44.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 22.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 32.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 76.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 50.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 52.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 267.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 73.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 128.T + 6.89e4T^{2} \) |
| 47 | \( 1 - 158.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 419.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 208.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 307.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 345.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 399.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 87.7T + 3.89e5T^{2} \) |
| 79 | \( 1 - 221.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 433.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 18.7T + 7.04e5T^{2} \) |
| 97 | \( 1 - 334.T + 9.12e5T^{2} \) |
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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
−8.449832686519377098391048573833, −7.911297801673892560850206270693, −7.43485761761297193712943216856, −6.57832695802347833421959374572, −4.98746199551536167166108316404, −4.02801199815976076739545803308, −3.27204403697343076990248005796, −2.66225724113714195116117766591, −0.824578656321021843626234606292, 0,
0.824578656321021843626234606292, 2.66225724113714195116117766591, 3.27204403697343076990248005796, 4.02801199815976076739545803308, 4.98746199551536167166108316404, 6.57832695802347833421959374572, 7.43485761761297193712943216856, 7.911297801673892560850206270693, 8.449832686519377098391048573833
