| L(s) = 1 | + 5.13·5-s + 4.02i·7-s + 4.30i·11-s + 18.4·13-s + 23.5·17-s − 21.7i·19-s − 30.7i·23-s + 1.34·25-s − 12.6·29-s − 24.5i·31-s + 20.6i·35-s − 18.2·37-s + 38.0·41-s + 34.9i·43-s + 29.6i·47-s + ⋯ |
| L(s) = 1 | + 1.02·5-s + 0.575i·7-s + 0.391i·11-s + 1.42·13-s + 1.38·17-s − 1.14i·19-s − 1.33i·23-s + 0.0536·25-s − 0.437·29-s − 0.791i·31-s + 0.590i·35-s − 0.492·37-s + 0.929·41-s + 0.813i·43-s + 0.630i·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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.860603284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.860603284\) |
| \(L(2)\) |
|
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 | \( 1 - 5.13T + 25T^{2} \) |
| 7 | \( 1 - 4.02iT - 49T^{2} \) |
| 11 | \( 1 - 4.30iT - 121T^{2} \) |
| 13 | \( 1 - 18.4T + 169T^{2} \) |
| 17 | \( 1 - 23.5T + 289T^{2} \) |
| 19 | \( 1 + 21.7iT - 361T^{2} \) |
| 23 | \( 1 + 30.7iT - 529T^{2} \) |
| 29 | \( 1 + 12.6T + 841T^{2} \) |
| 31 | \( 1 + 24.5iT - 961T^{2} \) |
| 37 | \( 1 + 18.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 38.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 34.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 29.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 39.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 65.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 29.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 11.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 140. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 119.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 9.18iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 113. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.88T + 7.92e3T^{2} \) |
| 97 | \( 1 - 55.5T + 9.40e3T^{2} \) |
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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
−9.236562121597200113105609066017, −8.462454279312951190086764849723, −7.60980972445605752962990370092, −6.49189595374886431817795780481, −5.95401640101779527348931775087, −5.22643469173148103713410034252, −4.15702007443495061760026190023, −2.98253527947236713919839635130, −2.08386496587734030294022076838, −0.951974211211188949844246594516,
1.05835827692874315216956013028, 1.81870989066385292734661651505, 3.36722437340851111026292329038, 3.84852536504579357956639565898, 5.44670641245091962848691981441, 5.71645802005359564525045221828, 6.63323919466320222114316730105, 7.62601700407406526095699243053, 8.324263694564595874969976446239, 9.244468982966529138721258817979
