| L(s) = 1 | + 2.35·3-s + 5.25·7-s − 3.47·9-s + 19.9i·11-s + 8.47i·13-s − 11.8i·17-s + 15.2i·19-s + 12.3·21-s − 0.555·23-s − 29.3·27-s − 10.9·29-s + 8.29i·31-s + 46.8i·33-s − 18.3i·37-s + 19.9i·39-s + ⋯ |
| L(s) = 1 | + 0.783·3-s + 0.751·7-s − 0.385·9-s + 1.81i·11-s + 0.651i·13-s − 0.699i·17-s + 0.800i·19-s + 0.588·21-s − 0.0241·23-s − 1.08·27-s − 0.377·29-s + 0.267i·31-s + 1.41i·33-s − 0.496i·37-s + 0.510i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.878238424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.878238424\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.35T + 9T^{2} \) |
| 7 | \( 1 - 5.25T + 49T^{2} \) |
| 11 | \( 1 - 19.9iT - 121T^{2} \) |
| 13 | \( 1 - 8.47iT - 169T^{2} \) |
| 17 | \( 1 + 11.8iT - 289T^{2} \) |
| 19 | \( 1 - 15.2iT - 361T^{2} \) |
| 23 | \( 1 + 0.555T + 529T^{2} \) |
| 29 | \( 1 + 10.9T + 841T^{2} \) |
| 31 | \( 1 - 8.29iT - 961T^{2} \) |
| 37 | \( 1 + 18.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 14.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 22.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 53.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 66.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 17.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 90.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 50.2T + 4.48e3T^{2} \) |
| 71 | \( 1 - 80.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.55iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 13.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 76.2T + 6.88e3T^{2} \) |
| 89 | \( 1 - 111.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 92.8iT - 9.40e3T^{2} \) |
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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
−9.430933313985729017577876835345, −8.716516618737483051077037754924, −7.81815719851711351864147383097, −7.35924976397323141845832817641, −6.35929299911892372482145933241, −5.15564162840893322742825917010, −4.50354744541008157817480958014, −3.49193555744582128817302331610, −2.29949431337894806661919448497, −1.63810502211790043735955232982,
0.42033328563307713636297379432, 1.80998078577950419824987656257, 3.02835371295334174874839175940, 3.53343844356000553909925507880, 4.84451330489238604805728883787, 5.69582104667361444646574298461, 6.45498357798977022742452642609, 7.73915171567713339495204739133, 8.291915104911985251908617177503, 8.689968676409976601190627876468
