| L(s) = 1 | + 4.50·2-s − 9.57·3-s + 12.3·4-s + 3.79·5-s − 43.1·6-s − 1.88·7-s + 19.4·8-s + 64.6·9-s + 17.1·10-s − 66.6·11-s − 117.·12-s + 12.0·13-s − 8.48·14-s − 36.3·15-s − 10.9·16-s − 54.8·17-s + 291.·18-s − 104.·19-s + 46.7·20-s + 18.0·21-s − 300.·22-s − 109.·23-s − 185.·24-s − 110.·25-s + 54.2·26-s − 359.·27-s − 23.1·28-s + ⋯ |
| L(s) = 1 | + 1.59·2-s − 1.84·3-s + 1.53·4-s + 0.339·5-s − 2.93·6-s − 0.101·7-s + 0.857·8-s + 2.39·9-s + 0.541·10-s − 1.82·11-s − 2.83·12-s + 0.256·13-s − 0.161·14-s − 0.625·15-s − 0.171·16-s − 0.782·17-s + 3.81·18-s − 1.26·19-s + 0.522·20-s + 0.187·21-s − 2.91·22-s − 0.992·23-s − 1.58·24-s − 0.884·25-s + 0.408·26-s − 2.56·27-s − 0.156·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 229 | \( 1 - 229T \) |
| good | 2 | \( 1 - 4.50T + 8T^{2} \) |
| 3 | \( 1 + 9.57T + 27T^{2} \) |
| 5 | \( 1 - 3.79T + 125T^{2} \) |
| 7 | \( 1 + 1.88T + 343T^{2} \) |
| 11 | \( 1 + 66.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 12.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 54.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 104.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 109.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 42.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 308.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 406.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 136.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 56.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 3.11T + 1.03e5T^{2} \) |
| 53 | \( 1 + 393.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 321.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 472.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 255.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 486.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 392.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 23.2T + 4.93e5T^{2} \) |
| 83 | \( 1 - 712.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.28e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 473.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.44099892518381462892576883911, −10.80696292249879761393117873561, −9.939438882585492529900094231026, −7.81005966889721565788038400029, −6.33360389109876803585638064467, −6.04041618962313849282245097790, −4.96883878488941582320335908321, −4.27675314795584213992639640618, −2.33242515397673045594515226228, 0,
2.33242515397673045594515226228, 4.27675314795584213992639640618, 4.96883878488941582320335908321, 6.04041618962313849282245097790, 6.33360389109876803585638064467, 7.81005966889721565788038400029, 9.939438882585492529900094231026, 10.80696292249879761393117873561, 11.44099892518381462892576883911
