L(s) = 1 | + (1.07 − 1.86i)2-s + (−0.635 − 1.61i)3-s + (−1.32 − 2.29i)4-s + (−1.81 − 3.13i)5-s + (−3.69 − 0.550i)6-s + (−1.13 + 1.96i)7-s − 1.39·8-s + (−2.19 + 2.04i)9-s − 7.81·10-s + (−2.85 + 3.58i)12-s + (0.619 + 1.07i)13-s + (2.44 + 4.23i)14-s + (−3.90 + 4.91i)15-s + (1.14 − 1.98i)16-s − 5.69·17-s + (1.45 + 6.30i)18-s + ⋯ |
L(s) = 1 | + (0.762 − 1.32i)2-s + (−0.366 − 0.930i)3-s + (−0.661 − 1.14i)4-s + (−0.810 − 1.40i)5-s + (−1.50 − 0.224i)6-s + (−0.429 + 0.743i)7-s − 0.492·8-s + (−0.730 + 0.682i)9-s − 2.47·10-s + (−0.823 + 1.03i)12-s + (0.171 + 0.297i)13-s + (0.654 + 1.13i)14-s + (−1.00 + 1.26i)15-s + (0.286 − 0.495i)16-s − 1.38·17-s + (0.343 + 1.48i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7884729834\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7884729834\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.635 + 1.61i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-1.07 + 1.86i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (1.81 + 3.13i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (1.13 - 1.96i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (-0.619 - 1.07i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 5.69T + 17T^{2} \) |
| 19 | \( 1 - 2.89T + 19T^{2} \) |
| 23 | \( 1 + (2.95 + 5.12i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.75 - 3.03i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.25 - 2.17i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 0.333T + 37T^{2} \) |
| 41 | \( 1 + (4.98 + 8.63i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.57 - 6.19i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.34 + 7.53i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 6.16T + 53T^{2} \) |
| 59 | \( 1 + (1.45 + 2.52i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.13 + 5.43i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.68 + 8.10i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 + 4.31T + 73T^{2} \) |
| 79 | \( 1 + (-0.708 + 1.22i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.37 - 2.38i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 4.77T + 89T^{2} \) |
| 97 | \( 1 + (-1.27 + 2.21i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.032304459367122000653844347907, −8.637930916555716131104553021466, −7.57622576363808160402325423204, −6.47465930081412613088703279488, −5.36964601929225158422067097940, −4.74984398654325028100104808857, −3.79383824727963333923575082374, −2.54603019679810592487863051104, −1.56986184068512073044556605156, −0.28270308342861646940142264859,
3.02866272517221715681387709476, 3.87710263832594609031470472515, 4.37446289942802884649815432536, 5.60829839601018812321205623936, 6.34412230257438824021878454448, 7.06755827546943209866320969912, 7.63922677025566226068369068520, 8.667105874661911955537529186052, 9.916916487151501211942797353844, 10.49852060780443322625715665656