| L(s) = 1 | − 2-s + 0.732·3-s + 4-s + 1.73·5-s − 0.732·6-s + 4·7-s − 8-s − 2.46·9-s − 1.73·10-s − 4.73·11-s + 0.732·12-s + 6·13-s − 4·14-s + 1.26·15-s + 16-s − 1.73·17-s + 2.46·18-s + 1.26·19-s + 1.73·20-s + 2.92·21-s + 4.73·22-s + 1.26·23-s − 0.732·24-s − 2.00·25-s − 6·26-s − 4·27-s + 4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.422·3-s + 0.5·4-s + 0.774·5-s − 0.298·6-s + 1.51·7-s − 0.353·8-s − 0.821·9-s − 0.547·10-s − 1.42·11-s + 0.211·12-s + 1.66·13-s − 1.06·14-s + 0.327·15-s + 0.250·16-s − 0.420·17-s + 0.580·18-s + 0.290·19-s + 0.387·20-s + 0.638·21-s + 1.00·22-s + 0.264·23-s − 0.149·24-s − 0.400·25-s − 1.17·26-s − 0.769·27-s + 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.011696947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.011696947\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 7 | \( 1 - 4T + 7T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 + 1.73T + 17T^{2} \) |
| 19 | \( 1 - 1.26T + 19T^{2} \) |
| 23 | \( 1 - 1.26T + 23T^{2} \) |
| 29 | \( 1 - 4.26T + 29T^{2} \) |
| 31 | \( 1 - 1.26T + 31T^{2} \) |
| 41 | \( 1 + 0.464T + 41T^{2} \) |
| 43 | \( 1 - 9.46T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 2.53T + 53T^{2} \) |
| 59 | \( 1 + 2.53T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 + 6.19T + 67T^{2} \) |
| 71 | \( 1 - 2.53T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 8.19T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 + 5.19T + 89T^{2} \) |
| 97 | \( 1 - 5.19T + 97T^{2} \) |
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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
−8.773951100717292849139924840591, −8.032916028100805766011187471310, −7.82977531527585309538076604026, −6.55473864654754237781971426335, −5.66918237008087793490332094782, −5.21737129514302875381338256246, −3.96988743623596276111019634378, −2.71225044833791298079279909190, −2.10434669388422248628612032688, −1.00284266146855375015938965620,
1.00284266146855375015938965620, 2.10434669388422248628612032688, 2.71225044833791298079279909190, 3.96988743623596276111019634378, 5.21737129514302875381338256246, 5.66918237008087793490332094782, 6.55473864654754237781971426335, 7.82977531527585309538076604026, 8.032916028100805766011187471310, 8.773951100717292849139924840591
