L(s) = 1 | + 2.05·2-s + (1.70 − 0.283i)3-s + 2.21·4-s + (−0.0731 − 0.126i)5-s + (3.50 − 0.582i)6-s + 0.446·8-s + (2.83 − 0.969i)9-s + (−0.150 − 0.260i)10-s + (−0.832 + 1.44i)11-s + (3.78 − 0.628i)12-s + (−0.0999 + 0.173i)13-s + (−0.160 − 0.195i)15-s − 3.51·16-s + (−3.13 − 5.43i)17-s + (5.83 − 1.99i)18-s + (−3.45 + 5.99i)19-s + ⋯ |
L(s) = 1 | + 1.45·2-s + (0.986 − 0.163i)3-s + 1.10·4-s + (−0.0327 − 0.0566i)5-s + (1.43 − 0.237i)6-s + 0.157·8-s + (0.946 − 0.323i)9-s + (−0.0474 − 0.0822i)10-s + (−0.250 + 0.434i)11-s + (1.09 − 0.181i)12-s + (−0.0277 + 0.0480i)13-s + (−0.0415 − 0.0505i)15-s − 0.879·16-s + (−0.760 − 1.31i)17-s + (1.37 − 0.469i)18-s + (−0.793 + 1.37i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.64735 - 0.268303i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.64735 - 0.268303i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.70 + 0.283i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 2.05T + 2T^{2} \) |
| 5 | \( 1 + (0.0731 + 0.126i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.832 - 1.44i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.0999 - 0.173i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.13 + 5.43i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.45 - 5.99i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.09 - 5.35i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.46 + 4.27i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 2.51T + 31T^{2} \) |
| 37 | \( 1 + (3.50 - 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.15 - 2.00i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.940 + 1.62i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 1.81T + 47T^{2} \) |
| 53 | \( 1 + (2.67 + 4.62i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 4.57T + 59T^{2} \) |
| 61 | \( 1 - 0.678T + 61T^{2} \) |
| 67 | \( 1 + 6.18T + 67T^{2} \) |
| 71 | \( 1 - 1.27T + 71T^{2} \) |
| 73 | \( 1 + (-0.778 - 1.34i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + (3.75 + 6.50i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (4.53 - 7.85i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.98 - 6.90i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.50352029200091292872760661673, −10.19079742436438029898394526330, −9.276881674252171908611991934834, −8.275612142936481435518437751810, −7.20269110808633851380625766160, −6.35489732423258101936188362701, −5.04508541911414919321160973490, −4.20752881099597758829632750848, −3.17312869262596026090421207065, −2.12273432553517660731884850959,
2.26319804427224440741408921516, 3.25453694760961284341379274111, 4.23595147968851732016820678304, 5.06613204638275187404922392982, 6.37457125916462719826121084635, 7.18213173188078507513822362249, 8.610206421645845529672189922353, 9.033663727860207314155962806984, 10.60117850138376024193883960700, 11.10118797415342173590516500736