L(s) = 1 | + 2.82i·2-s − 8.00·4-s + 44.3i·5-s − 22.6i·8-s − 125.·10-s + 72.1i·11-s + 119.·13-s + 64.0·16-s − 166. i·17-s + 146.·19-s − 355. i·20-s − 204.·22-s − 586. i·23-s − 1.34e3·25-s + 337. i·26-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.500·4-s + 1.77i·5-s − 0.353i·8-s − 1.25·10-s + 0.596i·11-s + 0.705·13-s + 0.250·16-s − 0.577i·17-s + 0.404·19-s − 0.887i·20-s − 0.421·22-s − 1.10i·23-s − 2.15·25-s + 0.498i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.5381610235\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5381610235\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2.82iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 44.3iT - 625T^{2} \) |
| 11 | \( 1 - 72.1iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 119.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 166. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 146.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 586. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 1.11e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.71e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 2.09e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.39e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 324.T + 3.41e6T^{2} \) |
| 47 | \( 1 + 1.73e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 2.80e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 5.42e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 1.58e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 1.05e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 4.24e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 5.14e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 8.61e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 4.62e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.21e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 9.81e3T + 8.85e7T^{2} \) |
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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
−9.600118185978069627776682103591, −8.462467961220340893956151574221, −7.53548295104583498723349947606, −6.88692899124476202266812678201, −6.30040968023340763411715864467, −5.28446596370527800462947003914, −4.01203305538462660320411539580, −3.13952449386003280525644381667, −2.01168557291929513251639228129, −0.12703997842131300392351915024,
1.09983075448834068412875668396, 1.70638827132145696275185000888, 3.41609639179540997523674410498, 4.12469584136477636580123970932, 5.34068558953964078631759475391, 5.65833962967598071411653336015, 7.29698907275831142737826870940, 8.358775805328489851585701261473, 8.918105428733958463737411807580, 9.414528600615544028228513885920