L(s) = 1 | + (2 + 2i)2-s + (−3.68 + 3.68i)3-s + 8i·4-s + (6.02 − 24.2i)5-s − 14.7·6-s + (−4.81 − 4.81i)7-s + (−16 + 16i)8-s + 53.7i·9-s + (60.5 − 36.4i)10-s − 104.·11-s + (−29.5 − 29.5i)12-s + (89.3 − 89.3i)13-s − 19.2i·14-s + (67.2 + 111. i)15-s − 64·16-s + (−180. − 180. i)17-s + ⋯ |
L(s) = 1 | + (0.5 + 0.5i)2-s + (−0.409 + 0.409i)3-s + 0.5i·4-s + (0.241 − 0.970i)5-s − 0.409·6-s + (−0.0982 − 0.0982i)7-s + (−0.250 + 0.250i)8-s + 0.663i·9-s + (0.605 − 0.364i)10-s − 0.860·11-s + (−0.204 − 0.204i)12-s + (0.528 − 0.528i)13-s − 0.0982i·14-s + (0.298 + 0.496i)15-s − 0.250·16-s + (−0.625 − 0.625i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.178215563\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.178215563\) |
\(L(3)\) |
|
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 | \( 1 + (-6.02 + 24.2i)T \) |
| 23 | \( 1 + (77.9 - 77.9i)T \) |
good | 3 | \( 1 + (3.68 - 3.68i)T - 81iT^{2} \) |
| 7 | \( 1 + (4.81 + 4.81i)T + 2.40e3iT^{2} \) |
| 11 | \( 1 + 104.T + 1.46e4T^{2} \) |
| 13 | \( 1 + (-89.3 + 89.3i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + (180. + 180. i)T + 8.35e4iT^{2} \) |
| 19 | \( 1 + 475. iT - 1.30e5T^{2} \) |
| 29 | \( 1 + 506. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 387.T + 9.23e5T^{2} \) |
| 37 | \( 1 + (1.01e3 + 1.01e3i)T + 1.87e6iT^{2} \) |
| 41 | \( 1 - 1.89e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (-1.14e3 + 1.14e3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + (-1.19e3 - 1.19e3i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + (-1.96e3 + 1.96e3i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 - 22.0iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 4.64e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + (2.93e3 + 2.93e3i)T + 2.01e7iT^{2} \) |
| 71 | \( 1 + 5.44e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (200. - 200. i)T - 2.83e7iT^{2} \) |
| 79 | \( 1 - 6.69e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (4.55e3 - 4.55e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + 1.21e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (-5.31e3 - 5.31e3i)T + 8.85e7iT^{2} \) |
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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
−11.36000322854927626595274331876, −10.50327733526263218578395674746, −9.294898575830065981948793524131, −8.316923988931871362352334146914, −7.31077056957257052283438776049, −5.84461007104108532874637828030, −5.10660868827684850748182123837, −4.26808832200248323675988832112, −2.48054006436761404260550406605, −0.34360730186364327114617943917,
1.55747104848854710757265106398, 2.92806112047841643994968894736, 4.10749411506210806382622758994, 5.80625420532822843255453588352, 6.35115063374554390947813990365, 7.50180874231003285243442801740, 8.961718485357188775335737171053, 10.19784331591801902701520629157, 10.82984738400867654060381926913, 11.79592883135225723173939559763