L(s) = 1 | − 1.86·2-s − 3-s + 1.46·4-s + 1.46·5-s + 1.86·6-s − 7-s + 8-s + 9-s − 2.72·10-s − 2.92·11-s − 1.46·12-s + 0.676·13-s + 1.86·14-s − 1.46·15-s − 4.78·16-s − 7.11·17-s − 1.86·18-s + 7.32·19-s + 2.13·20-s + 21-s + 5.44·22-s − 6.72·23-s − 24-s − 2.86·25-s − 1.25·26-s − 27-s − 1.46·28-s + ⋯ |
L(s) = 1 | − 1.31·2-s − 0.577·3-s + 0.731·4-s + 0.654·5-s + 0.759·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.860·10-s − 0.881·11-s − 0.422·12-s + 0.187·13-s + 0.497·14-s − 0.377·15-s − 1.19·16-s − 1.72·17-s − 0.438·18-s + 1.68·19-s + 0.478·20-s + 0.218·21-s + 1.16·22-s − 1.40·23-s − 0.204·24-s − 0.572·25-s − 0.246·26-s − 0.192·27-s − 0.276·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4774462211\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4774462211\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 - T \) |
good | 2 | \( 1 + 1.86T + 2T^{2} \) |
| 5 | \( 1 - 1.46T + 5T^{2} \) |
| 11 | \( 1 + 2.92T + 11T^{2} \) |
| 13 | \( 1 - 0.676T + 13T^{2} \) |
| 17 | \( 1 + 7.11T + 17T^{2} \) |
| 19 | \( 1 - 7.32T + 19T^{2} \) |
| 23 | \( 1 + 6.72T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 9.04T + 31T^{2} \) |
| 37 | \( 1 + 3.25T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 0.278T + 43T^{2} \) |
| 47 | \( 1 - 7.76T + 47T^{2} \) |
| 53 | \( 1 + 9.30T + 53T^{2} \) |
| 59 | \( 1 - 4.71T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 + 6.46T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 5.74T + 73T^{2} \) |
| 79 | \( 1 - 6.39T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 + 5.90T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−8.656806164389547171194062365515, −7.60460320017805967363235844058, −7.30855755687492464258846838049, −6.28249768223238983459308463717, −5.69128864789993793877536935360, −4.82429379188155609990241812197, −3.87244444709663126090545606047, −2.49199329038183555160502862064, −1.75441985654495871116973962895, −0.49059418597479321147403389026,
0.49059418597479321147403389026, 1.75441985654495871116973962895, 2.49199329038183555160502862064, 3.87244444709663126090545606047, 4.82429379188155609990241812197, 5.69128864789993793877536935360, 6.28249768223238983459308463717, 7.30855755687492464258846838049, 7.60460320017805967363235844058, 8.656806164389547171194062365515