| L(s) = 1 | + 5.40·2-s + 2.51·3-s + 21.2·4-s + 3.24·5-s + 13.5·6-s − 35.6·7-s + 71.4·8-s − 20.6·9-s + 17.5·10-s − 42.5·11-s + 53.3·12-s + 22.6·13-s − 192.·14-s + 8.17·15-s + 216.·16-s − 6.66·17-s − 111.·18-s − 14.7·19-s + 68.9·20-s − 89.5·21-s − 230.·22-s − 95.0·23-s + 179.·24-s − 114.·25-s + 122.·26-s − 119.·27-s − 755.·28-s + ⋯ |
| L(s) = 1 | + 1.91·2-s + 0.484·3-s + 2.65·4-s + 0.290·5-s + 0.925·6-s − 1.92·7-s + 3.15·8-s − 0.765·9-s + 0.555·10-s − 1.16·11-s + 1.28·12-s + 0.482·13-s − 3.67·14-s + 0.140·15-s + 3.38·16-s − 0.0950·17-s − 1.46·18-s − 0.178·19-s + 0.770·20-s − 0.930·21-s − 2.22·22-s − 0.861·23-s + 1.52·24-s − 0.915·25-s + 0.922·26-s − 0.854·27-s − 5.10·28-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 | \( 1 - 5.40T + 8T^{2} \) |
| 3 | \( 1 - 2.51T + 27T^{2} \) |
| 5 | \( 1 - 3.24T + 125T^{2} \) |
| 7 | \( 1 + 35.6T + 343T^{2} \) |
| 11 | \( 1 + 42.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 22.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 6.66T + 4.91e3T^{2} \) |
| 19 | \( 1 + 14.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 95.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 125.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 56.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 217.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 342.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 399.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 211.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 435.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 693.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 141.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 632.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 736.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 165.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.20e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 385.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 625.T + 9.12e5T^{2} \) |
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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
−8.271808375839147981883643238814, −7.38714657288410012220144055187, −6.51728179486602127869296061792, −5.86442246319795583270876646506, −5.46563527682729534069171199808, −4.11959429519266435676915967344, −3.43038712257641896314818966461, −2.80010407791379254787828911408, −2.09618136508929112060651125477, 0,
2.09618136508929112060651125477, 2.80010407791379254787828911408, 3.43038712257641896314818966461, 4.11959429519266435676915967344, 5.46563527682729534069171199808, 5.86442246319795583270876646506, 6.51728179486602127869296061792, 7.38714657288410012220144055187, 8.271808375839147981883643238814
