| L(s) = 1 | + 1.02·2-s − 1.11·3-s − 6.95·4-s − 17.8·5-s − 1.13·6-s − 19.7·7-s − 15.2·8-s − 25.7·9-s − 18.2·10-s + 22.7·11-s + 7.71·12-s + 19.7·13-s − 20.1·14-s + 19.7·15-s + 39.9·16-s − 101.·17-s − 26.3·18-s + 71.2·19-s + 123.·20-s + 21.9·21-s + 23.3·22-s + 39.5·23-s + 16.9·24-s + 192.·25-s + 20.2·26-s + 58.5·27-s + 137.·28-s + ⋯ |
| L(s) = 1 | + 0.361·2-s − 0.213·3-s − 0.869·4-s − 1.59·5-s − 0.0772·6-s − 1.06·7-s − 0.675·8-s − 0.954·9-s − 0.576·10-s + 0.624·11-s + 0.185·12-s + 0.421·13-s − 0.385·14-s + 0.340·15-s + 0.624·16-s − 1.45·17-s − 0.345·18-s + 0.860·19-s + 1.38·20-s + 0.227·21-s + 0.225·22-s + 0.358·23-s + 0.144·24-s + 1.54·25-s + 0.152·26-s + 0.417·27-s + 0.926·28-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 | \( 1 - 1.02T + 8T^{2} \) |
| 3 | \( 1 + 1.11T + 27T^{2} \) |
| 5 | \( 1 + 17.8T + 125T^{2} \) |
| 7 | \( 1 + 19.7T + 343T^{2} \) |
| 11 | \( 1 - 22.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 19.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 101.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 71.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 39.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 100.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 188.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 431.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 388.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 444.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 561.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 582.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 490.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 638.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 37.3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 504.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 690.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 112.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 772.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 403.T + 9.12e5T^{2} \) |
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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
−8.469985624780428215713053211406, −7.916287231030001891823479293051, −6.63801880472538996693589871060, −6.25974805708967487779435168595, −4.97055334576475624873965417739, −4.35507546024553913765716835271, −3.44677313350485098749546732297, −2.98099489657457279686552201482, −0.76586288639847085482345866285, 0,
0.76586288639847085482345866285, 2.98099489657457279686552201482, 3.44677313350485098749546732297, 4.35507546024553913765716835271, 4.97055334576475624873965417739, 6.25974805708967487779435168595, 6.63801880472538996693589871060, 7.916287231030001891823479293051, 8.469985624780428215713053211406
