L(s) = 1 | + (−0.723 + 1.84i)2-s + (0.680 + 1.59i)3-s + (−1.40 − 1.30i)4-s + (−3.94 + 1.90i)5-s + (−3.42 + 0.101i)6-s + (2.61 + 0.408i)7-s + (−0.143 + 0.0693i)8-s + (−2.07 + 2.16i)9-s + (−0.648 − 8.65i)10-s + (2.48 + 3.11i)11-s + (1.12 − 3.12i)12-s + (−0.678 + 1.72i)13-s + (−2.64 + 4.52i)14-s + (−5.71 − 4.99i)15-s + (−0.310 − 4.14i)16-s + (2.33 − 2.16i)17-s + ⋯ |
L(s) = 1 | + (−0.511 + 1.30i)2-s + (0.392 + 0.919i)3-s + (−0.703 − 0.652i)4-s + (−1.76 + 0.850i)5-s + (−1.39 + 0.0413i)6-s + (0.988 + 0.154i)7-s + (−0.0508 + 0.0245i)8-s + (−0.691 + 0.722i)9-s + (−0.204 − 2.73i)10-s + (0.749 + 0.940i)11-s + (0.324 − 0.903i)12-s + (−0.188 + 0.479i)13-s + (−0.706 + 1.20i)14-s + (−1.47 − 1.28i)15-s + (−0.0776 − 1.03i)16-s + (0.566 − 0.525i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.292 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.463477 - 0.626514i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.463477 - 0.626514i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.680 - 1.59i)T \) |
| 7 | \( 1 + (-2.61 - 0.408i)T \) |
good | 2 | \( 1 + (0.723 - 1.84i)T + (-1.46 - 1.36i)T^{2} \) |
| 5 | \( 1 + (3.94 - 1.90i)T + (3.11 - 3.90i)T^{2} \) |
| 11 | \( 1 + (-2.48 - 3.11i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (0.678 - 1.72i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (-2.33 + 2.16i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.202 - 0.350i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.50 + 6.58i)T + (-20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (2.69 + 0.832i)T + (23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (3.16 - 5.48i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.63 - 1.73i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (-3.76 - 2.56i)T + (14.9 + 38.1i)T^{2} \) |
| 43 | \( 1 + (6.79 - 4.63i)T + (15.7 - 40.0i)T^{2} \) |
| 47 | \( 1 + (2.65 - 6.75i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (3.75 - 1.15i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (-2.71 + 1.84i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (2.89 - 2.68i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-0.467 + 0.808i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.209 + 0.918i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (2.49 + 0.376i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-2.76 - 4.79i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.437 + 1.11i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (4.08 + 10.4i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (4.31 - 7.48i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.55524210330465298449774479791, −10.86130022391740491595699589260, −9.668377310590879307401931418130, −8.724729265448433345040019240187, −7.983441895811996229215955824629, −7.40898996715686215641052265037, −6.52691104250052448087498933763, −4.88785716252023858959040122366, −4.25097716919748057542821048578, −2.89779594310863030119237954056,
0.59681514759461082272037780038, 1.56855560719768128769264194190, 3.33939204486170859357490352756, 3.96626796917058599840609880238, 5.58524675753933882193038347374, 7.28396355277804509954583148485, 8.067138677672280584667409891739, 8.584651002177227606133578634945, 9.410426152987224463991832421725, 11.04976471721558107180592118615