L(s) = 1 | + (−2 + 2i)2-s + (6.39 + 6.39i)3-s − 8i·4-s + (22.1 − 11.5i)5-s − 25.5·6-s + (18.6 − 18.6i)7-s + (16 + 16i)8-s + 0.837i·9-s + (−21.2 + 67.4i)10-s + 60.8·11-s + (51.1 − 51.1i)12-s + (−52.7 − 52.7i)13-s + 74.7i·14-s + (215. + 68.0i)15-s − 64·16-s + (36.5 − 36.5i)17-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.5i)2-s + (0.710 + 0.710i)3-s − 0.5i·4-s + (0.887 − 0.461i)5-s − 0.710·6-s + (0.381 − 0.381i)7-s + (0.250 + 0.250i)8-s + 0.0103i·9-s + (−0.212 + 0.674i)10-s + 0.503·11-s + (0.355 − 0.355i)12-s + (−0.312 − 0.312i)13-s + 0.381i·14-s + (0.958 + 0.302i)15-s − 0.250·16-s + (0.126 − 0.126i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0738i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.997 + 0.0738i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.275367167\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.275367167\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2 - 2i)T \) |
| 5 | \( 1 + (-22.1 + 11.5i)T \) |
| 23 | \( 1 + (77.9 + 77.9i)T \) |
good | 3 | \( 1 + (-6.39 - 6.39i)T + 81iT^{2} \) |
| 7 | \( 1 + (-18.6 + 18.6i)T - 2.40e3iT^{2} \) |
| 11 | \( 1 - 60.8T + 1.46e4T^{2} \) |
| 13 | \( 1 + (52.7 + 52.7i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 + (-36.5 + 36.5i)T - 8.35e4iT^{2} \) |
| 19 | \( 1 + 628. iT - 1.30e5T^{2} \) |
| 29 | \( 1 + 209. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.45e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + (-810. + 810. i)T - 1.87e6iT^{2} \) |
| 41 | \( 1 + 10.7T + 2.82e6T^{2} \) |
| 43 | \( 1 + (-931. - 931. i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + (487. - 487. i)T - 4.87e6iT^{2} \) |
| 53 | \( 1 + (-1.52e3 - 1.52e3i)T + 7.89e6iT^{2} \) |
| 59 | \( 1 - 5.35e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 3.08e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + (-4.05e3 + 4.05e3i)T - 2.01e7iT^{2} \) |
| 71 | \( 1 - 1.06e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (1.81e3 + 1.81e3i)T + 2.83e7iT^{2} \) |
| 79 | \( 1 - 6.10e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (-7.07e3 - 7.07e3i)T + 4.74e7iT^{2} \) |
| 89 | \( 1 + 3.66e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (1.70e3 - 1.70e3i)T - 8.85e7iT^{2} \) |
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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
−11.20954865615699557891956730760, −10.22888149886155353603147585669, −9.255487886310247888092823311256, −8.983510710292787618212845044583, −7.69394979565958194045843021308, −6.52392567617682945287078592179, −5.25920637274453538593985384585, −4.19857295917859504429773079793, −2.51620402029249454863685741495, −0.874412686949120004758157720738,
1.57113778953199978610635191921, 2.19966788565504360784792569870, 3.56758960227579270894213725443, 5.44006490946078576587718419744, 6.73479024431920674412786063933, 7.74695352646464411320078126779, 8.640636399146519514353539817715, 9.588015469061556485748118147617, 10.46543673253119559877357112633, 11.54775505376571223568048517338