L(s) = 1 | + 1.73·2-s + (1.36 − 2.36i)3-s + 0.999·4-s + (0.866 − 1.5i)5-s + (2.36 − 4.09i)6-s − 1.73·8-s + (−2.23 − 3.86i)9-s + (1.49 − 2.59i)10-s + (−0.633 + 1.09i)11-s + (1.36 − 2.36i)12-s + (−3.59 − 0.232i)13-s + (−2.36 − 4.09i)15-s − 5·16-s + 7.73·17-s + (−3.86 − 6.69i)18-s + (1 + 1.73i)19-s + ⋯ |
L(s) = 1 | + 1.22·2-s + (0.788 − 1.36i)3-s + 0.499·4-s + (0.387 − 0.670i)5-s + (0.965 − 1.67i)6-s − 0.612·8-s + (−0.744 − 1.28i)9-s + (0.474 − 0.821i)10-s + (−0.191 + 0.331i)11-s + (0.394 − 0.683i)12-s + (−0.997 − 0.0643i)13-s + (−0.610 − 1.05i)15-s − 1.25·16-s + 1.87·17-s + (−0.911 − 1.57i)18-s + (0.229 + 0.397i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0662 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0662 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.27591 - 2.43201i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.27591 - 2.43201i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (3.59 + 0.232i)T \) |
good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 3 | \( 1 + (-1.36 + 2.36i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.866 + 1.5i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.633 - 1.09i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 - 7.73T + 17T^{2} \) |
| 19 | \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 4.73T + 23T^{2} \) |
| 29 | \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.09 - 3.63i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + (2.59 + 4.5i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.0980 + 0.169i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.46 + 11.1i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.96 - 8.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 7.26T + 59T^{2} \) |
| 61 | \( 1 + (2.40 + 4.16i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.09 + 5.36i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (1.59 + 2.76i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8.09 - 14.0i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 2.19T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 + (-3.19 + 5.53i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.32036587906443487761355879767, −9.265006094244007721464977665262, −8.525398062121784186851782908317, −7.47288819113341873011786654496, −6.84920898320166601920928986849, −5.54088018576110162034431313542, −5.04452056639603304805067437160, −3.50023209342669921399257422927, −2.62151928459227870422242844324, −1.30317478076645005608008485413,
2.80355517524657599539568721867, 3.11471394827621832236420945539, 4.27938418678846632208569759361, 5.07188701256020282566703243885, 5.87148668643395276411381107943, 7.13680809550509251332853021222, 8.308632071979103894335537542411, 9.349526150791865938471625242548, 9.929725635353955361085998757982, 10.66294013918152891218119157701