L(s) = 1 | + (−1.34 − 0.436i)2-s + (1.71 − 1.71i)3-s + (1.61 + 1.17i)4-s + (−0.923 − 2.03i)5-s + (−3.06 + 1.56i)6-s + (0.323 + 0.323i)7-s + (−1.66 − 2.28i)8-s − 2.90i·9-s + (0.353 + 3.14i)10-s − 0.665i·11-s + (4.80 − 0.766i)12-s + (−3.14 − 3.14i)13-s + (−0.293 − 0.575i)14-s + (−5.08 − 1.91i)15-s + (1.24 + 3.80i)16-s + (1.48 − 1.48i)17-s + ⋯ |
L(s) = 1 | + (−0.951 − 0.308i)2-s + (0.992 − 0.992i)3-s + (0.809 + 0.586i)4-s + (−0.412 − 0.910i)5-s + (−1.25 + 0.637i)6-s + (0.122 + 0.122i)7-s + (−0.589 − 0.807i)8-s − 0.969i·9-s + (0.111 + 0.993i)10-s − 0.200i·11-s + (1.38 − 0.221i)12-s + (−0.872 − 0.872i)13-s + (−0.0784 − 0.153i)14-s + (−1.31 − 0.494i)15-s + (0.311 + 0.950i)16-s + (0.359 − 0.359i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.685 + 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.685 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.419450 - 0.970755i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.419450 - 0.970755i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.34 + 0.436i)T \) |
| 5 | \( 1 + (0.923 + 2.03i)T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + (-1.71 + 1.71i)T - 3iT^{2} \) |
| 7 | \( 1 + (-0.323 - 0.323i)T + 7iT^{2} \) |
| 11 | \( 1 + 0.665iT - 11T^{2} \) |
| 13 | \( 1 + (3.14 + 3.14i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.48 + 1.48i)T - 17iT^{2} \) |
| 23 | \( 1 + (-4.04 + 4.04i)T - 23iT^{2} \) |
| 29 | \( 1 + 1.94iT - 29T^{2} \) |
| 31 | \( 1 - 7.94iT - 31T^{2} \) |
| 37 | \( 1 + (2.86 - 2.86i)T - 37iT^{2} \) |
| 41 | \( 1 - 0.378T + 41T^{2} \) |
| 43 | \( 1 + (-6.48 + 6.48i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.849 - 0.849i)T + 47iT^{2} \) |
| 53 | \( 1 + (7.26 + 7.26i)T + 53iT^{2} \) |
| 59 | \( 1 - 0.564T + 59T^{2} \) |
| 61 | \( 1 + 6.86T + 61T^{2} \) |
| 67 | \( 1 + (-11.3 - 11.3i)T + 67iT^{2} \) |
| 71 | \( 1 - 0.795iT - 71T^{2} \) |
| 73 | \( 1 + (-8.57 - 8.57i)T + 73iT^{2} \) |
| 79 | \( 1 - 9.50T + 79T^{2} \) |
| 83 | \( 1 + (-8.07 + 8.07i)T - 83iT^{2} \) |
| 89 | \( 1 - 16.8iT - 89T^{2} \) |
| 97 | \( 1 + (-2.53 + 2.53i)T - 97iT^{2} \) |
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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
−10.97795519794811891693098871903, −9.871867409020361767326949203514, −8.846525944279659233068559548799, −8.342818423283712563781259970319, −7.60492499255018696798793857599, −6.80134384378372818682049745070, −5.12862015164826496854544280901, −3.36799986051654236970471056752, −2.27801289344887496554151290649, −0.866483556488401313823870552673,
2.27035476406557126893935542623, 3.39886130279132125013491497582, 4.64781939414803707194353668549, 6.24379978914943153682916610368, 7.38968636705000367612314756962, 7.952595180408876689185082622565, 9.246755530515928174998418807718, 9.557163999039794788435797427291, 10.59304747350909254642247053905, 11.20324381749063769706093947700