L(s) = 1 | − 1.93i·2-s − 1.73·4-s − 5-s + (1 − 2.44i)7-s − 0.517i·8-s + 1.93i·10-s − 5.27i·11-s + 6.69i·13-s + (−4.73 − 1.93i)14-s − 4.46·16-s − 3.46·17-s − 6.69i·19-s + 1.73·20-s − 10.1·22-s + 1.41i·23-s + ⋯ |
L(s) = 1 | − 1.36i·2-s − 0.866·4-s − 0.447·5-s + (0.377 − 0.925i)7-s − 0.183i·8-s + 0.610i·10-s − 1.59i·11-s + 1.85i·13-s + (−1.26 − 0.516i)14-s − 1.11·16-s − 0.840·17-s − 1.53i·19-s + 0.387·20-s − 2.17·22-s + 0.294i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.129829 - 1.13450i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.129829 - 1.13450i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-1 + 2.44i)T \) |
good | 2 | \( 1 + 1.93iT - 2T^{2} \) |
| 11 | \( 1 + 5.27iT - 11T^{2} \) |
| 13 | \( 1 - 6.69iT - 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 6.69iT - 19T^{2} \) |
| 23 | \( 1 - 1.41iT - 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 + 1.79iT - 31T^{2} \) |
| 37 | \( 1 - 5.46T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 8.92T + 43T^{2} \) |
| 47 | \( 1 - 2.53T + 47T^{2} \) |
| 53 | \( 1 - 4.52iT - 53T^{2} \) |
| 59 | \( 1 - 2.53T + 59T^{2} \) |
| 61 | \( 1 + 3.58iT - 61T^{2} \) |
| 67 | \( 1 - 2.92T + 67T^{2} \) |
| 71 | \( 1 - 9.41iT - 71T^{2} \) |
| 73 | \( 1 + 6.69iT - 73T^{2} \) |
| 79 | \( 1 - 2.92T + 79T^{2} \) |
| 83 | \( 1 + 2.53T + 83T^{2} \) |
| 89 | \( 1 + 0.928T + 89T^{2} \) |
| 97 | \( 1 - 10.2iT - 97T^{2} \) |
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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
−11.12377364491127807752431291750, −10.89990576946031219567400863911, −9.399118480375941869621998720136, −8.814570820759802067482863128329, −7.38450281613412684332292429467, −6.42599872617357549692423868531, −4.53115895104720968665450312972, −3.87004547748875191951105188808, −2.50460288368402098040666663796, −0.854850235103496646194412884209,
2.41677867244670426775587416637, 4.35825056307535366849534896428, 5.39064607497294255953299002215, 6.19491046902529440168830119634, 7.51398810241649554935762631301, 7.941245228743110013209012295908, 8.912688762507248293230162214516, 10.06713061858503804127670437109, 11.18014970939378780546480439397, 12.36265677901363705702187679190