| L(s) = 1 | + 2.26·2-s + 3.17·3-s + 3.12·4-s + 2.32·5-s + 7.19·6-s − 4.39·7-s + 2.55·8-s + 7.08·9-s + 5.27·10-s + 11-s + 9.94·12-s − 9.95·14-s + 7.39·15-s − 0.464·16-s − 3.60·17-s + 16.0·18-s − 2.34·19-s + 7.28·20-s − 13.9·21-s + 2.26·22-s + 2.46·23-s + 8.12·24-s + 0.417·25-s + 12.9·27-s − 13.7·28-s + 6.42·29-s + 16.7·30-s + ⋯ |
| L(s) = 1 | + 1.60·2-s + 1.83·3-s + 1.56·4-s + 1.04·5-s + 2.93·6-s − 1.66·7-s + 0.904·8-s + 2.36·9-s + 1.66·10-s + 0.301·11-s + 2.86·12-s − 2.66·14-s + 1.90·15-s − 0.116·16-s − 0.874·17-s + 3.78·18-s − 0.537·19-s + 1.62·20-s − 3.04·21-s + 0.482·22-s + 0.513·23-s + 1.65·24-s + 0.0834·25-s + 2.49·27-s − 2.59·28-s + 1.19·29-s + 3.05·30-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
| \(L(1)\) |
\(\approx\) |
\(7.881013742\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.881013742\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.26T + 2T^{2} \) |
| 3 | \( 1 - 3.17T + 3T^{2} \) |
| 5 | \( 1 - 2.32T + 5T^{2} \) |
| 7 | \( 1 + 4.39T + 7T^{2} \) |
| 17 | \( 1 + 3.60T + 17T^{2} \) |
| 19 | \( 1 + 2.34T + 19T^{2} \) |
| 23 | \( 1 - 2.46T + 23T^{2} \) |
| 29 | \( 1 - 6.42T + 29T^{2} \) |
| 31 | \( 1 + 4.49T + 31T^{2} \) |
| 37 | \( 1 - 3.91T + 37T^{2} \) |
| 41 | \( 1 + 9.00T + 41T^{2} \) |
| 43 | \( 1 + 0.179T + 43T^{2} \) |
| 47 | \( 1 - 0.0336T + 47T^{2} \) |
| 53 | \( 1 + 7.80T + 53T^{2} \) |
| 59 | \( 1 - 7.69T + 59T^{2} \) |
| 61 | \( 1 - 3.02T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 - 7.10T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 - 2.77T + 79T^{2} \) |
| 83 | \( 1 - 9.48T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 - 5.83T + 97T^{2} \) |
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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
−9.165830595913795318585109043623, −8.721565908657433778237865308886, −7.38082143346850849010942856012, −6.53579548436755701968464492543, −6.25318417514342923978993934172, −4.94654693679836415528317172224, −4.00464087820973241610439108464, −3.30663545493914646704667526870, −2.66891451352066779781268349571, −1.93236350491908680747321227457,
1.93236350491908680747321227457, 2.66891451352066779781268349571, 3.30663545493914646704667526870, 4.00464087820973241610439108464, 4.94654693679836415528317172224, 6.25318417514342923978993934172, 6.53579548436755701968464492543, 7.38082143346850849010942856012, 8.721565908657433778237865308886, 9.165830595913795318585109043623
