| L(s) = 1 | + 2.55·2-s − 2.99·3-s + 4.52·4-s + 0.244·5-s − 7.65·6-s + 4.42·7-s + 6.44·8-s + 5.98·9-s + 0.625·10-s − 11-s − 13.5·12-s − 5.89·13-s + 11.2·14-s − 0.734·15-s + 7.41·16-s − 2.93·17-s + 15.2·18-s + 19-s + 1.10·20-s − 13.2·21-s − 2.55·22-s − 0.372·23-s − 19.3·24-s − 4.94·25-s − 15.0·26-s − 8.96·27-s + 20.0·28-s + ⋯ |
| L(s) = 1 | + 1.80·2-s − 1.73·3-s + 2.26·4-s + 0.109·5-s − 3.12·6-s + 1.67·7-s + 2.27·8-s + 1.99·9-s + 0.197·10-s − 0.301·11-s − 3.91·12-s − 1.63·13-s + 3.01·14-s − 0.189·15-s + 1.85·16-s − 0.712·17-s + 3.60·18-s + 0.229·19-s + 0.247·20-s − 2.89·21-s − 0.544·22-s − 0.0777·23-s − 3.94·24-s − 0.988·25-s − 2.95·26-s − 1.72·27-s + 3.78·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.169824296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.169824296\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.55T + 2T^{2} \) |
| 3 | \( 1 + 2.99T + 3T^{2} \) |
| 5 | \( 1 - 0.244T + 5T^{2} \) |
| 7 | \( 1 - 4.42T + 7T^{2} \) |
| 13 | \( 1 + 5.89T + 13T^{2} \) |
| 17 | \( 1 + 2.93T + 17T^{2} \) |
| 23 | \( 1 + 0.372T + 23T^{2} \) |
| 29 | \( 1 + 3.47T + 29T^{2} \) |
| 31 | \( 1 - 6.37T + 31T^{2} \) |
| 37 | \( 1 - 0.926T + 37T^{2} \) |
| 41 | \( 1 + 6.67T + 41T^{2} \) |
| 43 | \( 1 - 2.12T + 43T^{2} \) |
| 47 | \( 1 + 1.72T + 47T^{2} \) |
| 53 | \( 1 + 1.44T + 53T^{2} \) |
| 59 | \( 1 - 7.71T + 59T^{2} \) |
| 61 | \( 1 - 4.16T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 2.40T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 - 1.67T + 79T^{2} \) |
| 83 | \( 1 + 6.47T + 83T^{2} \) |
| 89 | \( 1 + 4.95T + 89T^{2} \) |
| 97 | \( 1 - 8.24T + 97T^{2} \) |
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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
−12.16374944357878071848301360135, −11.68839088475414473733057959434, −11.09103623651554728233725284728, −10.11325149907049710592478905561, −7.72658915817545839439684493353, −6.80248782016642489332906260128, −5.62973329720563020305381508200, −4.97883523731470481734744245001, −4.38511610735756297537590625805, −2.06675571676242139840767917581,
2.06675571676242139840767917581, 4.38511610735756297537590625805, 4.97883523731470481734744245001, 5.62973329720563020305381508200, 6.80248782016642489332906260128, 7.72658915817545839439684493353, 10.11325149907049710592478905561, 11.09103623651554728233725284728, 11.68839088475414473733057959434, 12.16374944357878071848301360135
