| L(s) = 1 | − 0.956·5-s − 6.34i·7-s + 14.4i·11-s − 1.76·13-s + 1.34·17-s − 13.0i·19-s − 4.19i·23-s − 24.0·25-s + 38.1·29-s + 0.172i·31-s + 6.07i·35-s − 15.1·37-s − 69.3·41-s + 5.52i·43-s − 66.7i·47-s + ⋯ |
| L(s) = 1 | − 0.191·5-s − 0.906i·7-s + 1.31i·11-s − 0.135·13-s + 0.0790·17-s − 0.685i·19-s − 0.182i·23-s − 0.963·25-s + 1.31·29-s + 0.00557i·31-s + 0.173i·35-s − 0.410·37-s − 1.69·41-s + 0.128i·43-s − 1.41i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1473573169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1473573169\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 0.956T + 25T^{2} \) |
| 7 | \( 1 + 6.34iT - 49T^{2} \) |
| 11 | \( 1 - 14.4iT - 121T^{2} \) |
| 13 | \( 1 + 1.76T + 169T^{2} \) |
| 17 | \( 1 - 1.34T + 289T^{2} \) |
| 19 | \( 1 + 13.0iT - 361T^{2} \) |
| 23 | \( 1 + 4.19iT - 529T^{2} \) |
| 29 | \( 1 - 38.1T + 841T^{2} \) |
| 31 | \( 1 - 0.172iT - 961T^{2} \) |
| 37 | \( 1 + 15.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + 69.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 5.52iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 66.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 31.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 12.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 92.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 26.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 85.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 103.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 68.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 137. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 67.3T + 7.92e3T^{2} \) |
| 97 | \( 1 + 95.6T + 9.40e3T^{2} \) |
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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
−8.701236289936737108616664217654, −7.86238924260280675803285068238, −7.08382367386383395720375154466, −6.63070332037289310660297056354, −5.30918191529962215774261142743, −4.52487660596404105893038771969, −3.80763850091956630491590850470, −2.59187633604016889560388280094, −1.43042810732465880947062999073, −0.03879565427560685921012873050,
1.45796478984853238768485067938, 2.73950507979496945166901248542, 3.50827926572362293592985538428, 4.62624338458929842152821164303, 5.69247247481693741523917145961, 6.08464814558111331908702638833, 7.17775409123994246290886752517, 8.219207043696273446920738855434, 8.559038225148615697020844943233, 9.461187926914897193046999877790
