| L(s) = 1 | − 1.70·3-s + (0.539 + 2.17i)5-s + 2.63i·7-s − 0.0783·9-s + 5.41i·11-s + 6.34·13-s + (−0.921 − 3.70i)15-s − 3.41i·17-s + 3.26i·19-s − 4.49i·21-s − 1.36i·23-s + (−4.41 + 2.34i)25-s + 5.26·27-s + 2i·29-s − 4.68·31-s + ⋯ |
| L(s) = 1 | − 0.986·3-s + (0.241 + 0.970i)5-s + 0.994i·7-s − 0.0261·9-s + 1.63i·11-s + 1.75·13-s + (−0.237 − 0.957i)15-s − 0.829i·17-s + 0.748i·19-s − 0.981i·21-s − 0.285i·23-s + (−0.883 + 0.468i)25-s + 1.01·27-s + 0.371i·29-s − 0.840·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 - 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.856 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9593597254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9593597254\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.539 - 2.17i)T \) |
| good | 3 | \( 1 + 1.70T + 3T^{2} \) |
| 7 | \( 1 - 2.63iT - 7T^{2} \) |
| 11 | \( 1 - 5.41iT - 11T^{2} \) |
| 13 | \( 1 - 6.34T + 13T^{2} \) |
| 17 | \( 1 + 3.41iT - 17T^{2} \) |
| 19 | \( 1 - 3.26iT - 19T^{2} \) |
| 23 | \( 1 + 1.36iT - 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 4.68T + 31T^{2} \) |
| 37 | \( 1 + 5.75T + 37T^{2} \) |
| 41 | \( 1 - 7.75T + 41T^{2} \) |
| 43 | \( 1 + 4.44T + 43T^{2} \) |
| 47 | \( 1 + 4.78iT - 47T^{2} \) |
| 53 | \( 1 + 1.65T + 53T^{2} \) |
| 59 | \( 1 - 3.26iT - 59T^{2} \) |
| 61 | \( 1 - 2.49iT - 61T^{2} \) |
| 67 | \( 1 + 7.86T + 67T^{2} \) |
| 71 | \( 1 - 6.15T + 71T^{2} \) |
| 73 | \( 1 - 13.5iT - 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 + 8.52T + 89T^{2} \) |
| 97 | \( 1 - 4.58iT - 97T^{2} \) |
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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
−10.17274312035707782528200403762, −9.286488344667673022082541858647, −8.458156545658472662624953554239, −7.28530062974327606077447128145, −6.59881019798553768588439200024, −5.82644615441645429882859907585, −5.25517334931006358123513035476, −3.99378935399928321483722585588, −2.81052550495562783546795003995, −1.69690758459532436636789097752,
0.50006639441712017858936159212, 1.35341684850600770582161312430, 3.37839948092828449855903856871, 4.17711065949868828021311528243, 5.29875706013866727866267658297, 5.97636255549516212114227388107, 6.48461187434482822695228356525, 7.85185903990327598520191288054, 8.596412394067439029563688303707, 9.154913895360889818094073524155
