| L(s) = 1 | − 0.445·2-s − 1.24·3-s − 1.80·4-s + 0.692·5-s + 0.554·6-s + 0.356·7-s + 1.69·8-s − 1.44·9-s − 0.307·10-s + 4.93·11-s + 2.24·12-s − 5.65·13-s − 0.158·14-s − 0.862·15-s + 2.85·16-s + 4.49·17-s + 0.643·18-s − 2.35·19-s − 1.24·20-s − 0.445·21-s − 2.19·22-s + 2.29·23-s − 2.10·24-s − 4.52·25-s + 2.51·26-s + 5.54·27-s − 0.643·28-s + ⋯ |
| L(s) = 1 | − 0.314·2-s − 0.719·3-s − 0.900·4-s + 0.309·5-s + 0.226·6-s + 0.134·7-s + 0.598·8-s − 0.481·9-s − 0.0973·10-s + 1.48·11-s + 0.648·12-s − 1.56·13-s − 0.0424·14-s − 0.222·15-s + 0.712·16-s + 1.08·17-s + 0.151·18-s − 0.540·19-s − 0.278·20-s − 0.0971·21-s − 0.468·22-s + 0.478·23-s − 0.430·24-s − 0.904·25-s + 0.493·26-s + 1.06·27-s − 0.121·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 \) |
| good | 2 | \( 1 + 0.445T + 2T^{2} \) |
| 3 | \( 1 + 1.24T + 3T^{2} \) |
| 5 | \( 1 - 0.692T + 5T^{2} \) |
| 7 | \( 1 - 0.356T + 7T^{2} \) |
| 11 | \( 1 - 4.93T + 11T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 - 4.49T + 17T^{2} \) |
| 19 | \( 1 + 2.35T + 19T^{2} \) |
| 23 | \( 1 - 2.29T + 23T^{2} \) |
| 31 | \( 1 + 6.69T + 31T^{2} \) |
| 37 | \( 1 + 4.93T + 37T^{2} \) |
| 41 | \( 1 - 3.10T + 41T^{2} \) |
| 43 | \( 1 + 3.40T + 43T^{2} \) |
| 47 | \( 1 + 6.44T + 47T^{2} \) |
| 53 | \( 1 + 4.69T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 + 1.64T + 61T^{2} \) |
| 67 | \( 1 + 2.32T + 67T^{2} \) |
| 71 | \( 1 + 7.33T + 71T^{2} \) |
| 73 | \( 1 - 5.62T + 73T^{2} \) |
| 79 | \( 1 + 4.66T + 79T^{2} \) |
| 83 | \( 1 - 4.45T + 83T^{2} \) |
| 89 | \( 1 + 5.67T + 89T^{2} \) |
| 97 | \( 1 + 0.180T + 97T^{2} \) |
| 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
−9.634571426092388048678734328271, −9.180186628378244183271390984288, −8.162964463067174675005997241239, −7.23128557661409761097530544170, −6.16916301824279904719825176654, −5.30258749551361911898967738788, −4.55850750782646412732879996716, −3.35950213102342294287071945153, −1.58168237263676207010410443459, 0,
1.58168237263676207010410443459, 3.35950213102342294287071945153, 4.55850750782646412732879996716, 5.30258749551361911898967738788, 6.16916301824279904719825176654, 7.23128557661409761097530544170, 8.162964463067174675005997241239, 9.180186628378244183271390984288, 9.634571426092388048678734328271
