L(s) = 1 | + (−0.533 − 1.30i)2-s + (−1.43 + 1.39i)4-s + (0.895 − 1.55i)5-s + (2.08 − 1.20i)7-s + (2.59 + 1.12i)8-s + (−2.50 − 0.345i)10-s + (1.36 − 0.790i)11-s + (−5.35 − 3.09i)13-s + (−2.69 − 2.09i)14-s + (0.0944 − 3.99i)16-s − 3.69i·17-s + 3.12·19-s + (0.886 + 3.47i)20-s + (−1.76 − 1.37i)22-s + (−1.36 + 2.35i)23-s + ⋯ |
L(s) = 1 | + (−0.377 − 0.926i)2-s + (−0.715 + 0.698i)4-s + (0.400 − 0.693i)5-s + (0.789 − 0.455i)7-s + (0.916 + 0.398i)8-s + (−0.793 − 0.109i)10-s + (0.412 − 0.238i)11-s + (−1.48 − 0.857i)13-s + (−0.719 − 0.558i)14-s + (0.0236 − 0.999i)16-s − 0.897i·17-s + 0.717·19-s + (0.198 + 0.775i)20-s + (−0.376 − 0.292i)22-s + (−0.283 + 0.491i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.272 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.272 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.624494 - 0.825996i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.624494 - 0.825996i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.533 + 1.30i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.895 + 1.55i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-2.08 + 1.20i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.36 + 0.790i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (5.35 + 3.09i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 3.69iT - 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 23 | \( 1 + (1.36 - 2.35i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.55 + 4.42i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-5.95 - 3.43i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 5.24iT - 37T^{2} \) |
| 41 | \( 1 + (-5.32 - 3.07i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.452 + 0.783i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.88 - 8.46i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 7.05T + 53T^{2} \) |
| 59 | \( 1 + (-6.10 - 3.52i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.05 + 1.76i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.03 + 1.79i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 3.31T + 71T^{2} \) |
| 73 | \( 1 - 0.631T + 73T^{2} \) |
| 79 | \( 1 + (7.82 - 4.51i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (13.5 - 7.82i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 1.16iT - 89T^{2} \) |
| 97 | \( 1 + (6.72 + 11.6i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.92823968356120620954880945925, −11.16876772823278043211189918286, −9.939954483767203882441934830454, −9.408187452077442006278635942435, −8.158924696422848560605379443541, −7.38313544566865382815504701155, −5.32291837280508005077978572264, −4.50001078673342003453756379657, −2.81526140886639181793701954804, −1.13438515222288973379094528550,
2.08995754414363346316245817379, 4.35460440352756803761129179064, 5.48528816360593129121393691329, 6.62927276971034451225420542784, 7.46469983198688237970084764829, 8.584529862640413941693577531779, 9.585101515177383318675192098297, 10.38556370678407448954270539129, 11.56694827031982629562613538336, 12.64051414713437197540603183031