| L(s) = 1 | + 0.263·2-s + 1.72·3-s − 1.93·4-s − 0.976·5-s + 0.455·6-s + 0.775·7-s − 1.03·8-s − 0.00797·9-s − 0.257·10-s + 11-s − 3.33·12-s + 0.204·14-s − 1.68·15-s + 3.58·16-s − 4.83·17-s − 0.00210·18-s + 0.373·19-s + 1.88·20-s + 1.34·21-s + 0.263·22-s + 0.0494·23-s − 1.79·24-s − 4.04·25-s − 5.20·27-s − 1.49·28-s − 0.390·29-s − 0.445·30-s + ⋯ |
| L(s) = 1 | + 0.186·2-s + 0.998·3-s − 0.965·4-s − 0.436·5-s + 0.186·6-s + 0.293·7-s − 0.366·8-s − 0.00265·9-s − 0.0813·10-s + 0.301·11-s − 0.964·12-s + 0.0546·14-s − 0.436·15-s + 0.897·16-s − 1.17·17-s − 0.000495·18-s + 0.0857·19-s + 0.421·20-s + 0.292·21-s + 0.0561·22-s + 0.0103·23-s − 0.365·24-s − 0.809·25-s − 1.00·27-s − 0.282·28-s − 0.0724·29-s − 0.0812·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.263T + 2T^{2} \) |
| 3 | \( 1 - 1.72T + 3T^{2} \) |
| 5 | \( 1 + 0.976T + 5T^{2} \) |
| 7 | \( 1 - 0.775T + 7T^{2} \) |
| 17 | \( 1 + 4.83T + 17T^{2} \) |
| 19 | \( 1 - 0.373T + 19T^{2} \) |
| 23 | \( 1 - 0.0494T + 23T^{2} \) |
| 29 | \( 1 + 0.390T + 29T^{2} \) |
| 31 | \( 1 + 2.30T + 31T^{2} \) |
| 37 | \( 1 - 3.31T + 37T^{2} \) |
| 41 | \( 1 + 2.78T + 41T^{2} \) |
| 43 | \( 1 + 8.64T + 43T^{2} \) |
| 47 | \( 1 + 7.18T + 47T^{2} \) |
| 53 | \( 1 + 3.42T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 - 1.31T + 61T^{2} \) |
| 67 | \( 1 + 0.0855T + 67T^{2} \) |
| 71 | \( 1 + 5.84T + 71T^{2} \) |
| 73 | \( 1 + 7.94T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 + 4.17T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
| 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
−8.687089931295633699168000508778, −8.308800982911645805731761152443, −7.54990801038078150774031591671, −6.48521332796202531496495995506, −5.44076715671487226233896971772, −4.52406067560178955361276506398, −3.82074452770888723899803590993, −3.02808265222719690875499254064, −1.79371499197990418078617646451, 0,
1.79371499197990418078617646451, 3.02808265222719690875499254064, 3.82074452770888723899803590993, 4.52406067560178955361276506398, 5.44076715671487226233896971772, 6.48521332796202531496495995506, 7.54990801038078150774031591671, 8.308800982911645805731761152443, 8.687089931295633699168000508778
