L(s) = 1 | + (−1.23 + 0.683i)2-s + (−0.203 + 0.0662i)3-s + (1.06 − 1.69i)4-s + (−1.71 − 1.43i)5-s + (0.207 − 0.221i)6-s + (0.740 − 0.740i)7-s + (−0.165 + 2.82i)8-s + (−2.38 + 1.73i)9-s + (3.10 + 0.611i)10-s + (−3.18 + 0.504i)11-s + (−0.105 + 0.415i)12-s + (0.927 − 0.673i)13-s + (−0.410 + 1.42i)14-s + (0.444 + 0.179i)15-s + (−1.72 − 3.60i)16-s + (3.08 + 6.06i)17-s + ⋯ |
L(s) = 1 | + (−0.875 + 0.482i)2-s + (−0.117 + 0.0382i)3-s + (0.533 − 0.845i)4-s + (−0.765 − 0.643i)5-s + (0.0846 − 0.0903i)6-s + (0.279 − 0.279i)7-s + (−0.0586 + 0.998i)8-s + (−0.796 + 0.578i)9-s + (0.981 + 0.193i)10-s + (−0.960 + 0.152i)11-s + (−0.0304 + 0.119i)12-s + (0.257 − 0.186i)13-s + (−0.109 + 0.380i)14-s + (0.114 + 0.0464i)15-s + (−0.430 − 0.902i)16-s + (0.748 + 1.46i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.665 - 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.157187 + 0.351001i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.157187 + 0.351001i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.23 - 0.683i)T \) |
| 5 | \( 1 + (1.71 + 1.43i)T \) |
good | 3 | \( 1 + (0.203 - 0.0662i)T + (2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 + (-0.740 + 0.740i)T - 7iT^{2} \) |
| 11 | \( 1 + (3.18 - 0.504i)T + (10.4 - 3.39i)T^{2} \) |
| 13 | \( 1 + (-0.927 + 0.673i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-3.08 - 6.06i)T + (-9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-1.53 - 3.01i)T + (-11.1 + 15.3i)T^{2} \) |
| 23 | \( 1 + (-1.28 - 8.13i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (1.07 - 2.10i)T + (-17.0 - 23.4i)T^{2} \) |
| 31 | \( 1 + (8.74 + 2.84i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.943 + 0.685i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (5.17 + 7.12i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 5.67T + 43T^{2} \) |
| 47 | \( 1 + (1.88 - 3.69i)T + (-27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-4.22 + 1.37i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (14.6 + 2.31i)T + (56.1 + 18.2i)T^{2} \) |
| 61 | \( 1 + (-13.0 + 2.06i)T + (58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (1.39 - 4.30i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-0.332 - 1.02i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-3.05 + 0.483i)T + (69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (-3.62 - 11.1i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (6.90 + 2.24i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-13.0 - 9.50i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (12.4 + 6.33i)T + (57.0 + 78.4i)T^{2} \) |
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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
−11.24986178100323654365394214033, −10.75260335745127167205169961606, −9.720901390004576062588884218319, −8.588973500348938533729783847171, −7.934463609413990153807913559946, −7.41221032710840788373681180052, −5.70455688169973380297212112417, −5.23785656976994997174984919040, −3.55634395634767896272677940573, −1.61052676767148412946757963643,
0.33641838689042174869088363124, 2.62158752335886302459319502146, 3.37660393913411158791495194784, 5.03080506431202728841337048868, 6.52222471565364609365537428603, 7.38623448461541357912113871780, 8.283727346097529308997069797995, 9.041890638547056743078258129577, 10.15443482881966651177079686561, 11.03830406348381250312471008318