L(s) = 1 | + (−0.707 − 1.22i)2-s + 2.82·3-s + (−0.999 + 1.73i)4-s + 2.23·5-s + (−2.00 − 3.46i)6-s − 3.87i·7-s + 2.82·8-s + 5.00·9-s + (−1.58 − 2.73i)10-s + (−2.82 + 4.89i)12-s + 4.89i·13-s + (−4.74 + 2.73i)14-s + 6.32·15-s + (−2.00 − 3.46i)16-s − 2.23·17-s + (−3.53 − 6.12i)18-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.866i)2-s + 1.63·3-s + (−0.499 + 0.866i)4-s + 0.999·5-s + (−0.816 − 1.41i)6-s − 1.46i·7-s + 0.999·8-s + 1.66·9-s + (−0.500 − 0.866i)10-s + (−0.816 + 1.41i)12-s + 1.35i·13-s + (−1.26 + 0.731i)14-s + 1.63·15-s + (−0.500 − 0.866i)16-s − 0.542·17-s + (−0.833 − 1.44i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.69308 - 1.10245i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.69308 - 1.10245i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 1.22i)T \) |
| 5 | \( 1 - 2.23T \) |
| 23 | \( 1 + (2.82 - 3.87i)T \) |
good | 3 | \( 1 - 2.82T + 3T^{2} \) |
| 7 | \( 1 + 3.87iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4.89iT - 13T^{2} \) |
| 17 | \( 1 + 2.23T + 17T^{2} \) |
| 19 | \( 1 + 6.32T + 19T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 1.73iT - 31T^{2} \) |
| 37 | \( 1 - 6.70T + 37T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 + 7.74iT - 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 + 2.23T + 53T^{2} \) |
| 59 | \( 1 + 8.66iT - 59T^{2} \) |
| 61 | \( 1 - 10.9iT - 61T^{2} \) |
| 67 | \( 1 - 11.6iT - 67T^{2} \) |
| 71 | \( 1 + 1.73iT - 71T^{2} \) |
| 73 | \( 1 - 4.89iT - 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 + 3.87iT - 83T^{2} \) |
| 89 | \( 1 + 10.9iT - 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−10.46597601762559580100677454668, −9.993658655222342013147910913289, −9.112442586145695567891693155649, −8.551287326858889008667136254639, −7.47527881683040656309734554695, −6.67207908645192803570515128508, −4.42117294655091432941247936253, −3.81856341845175846131875470047, −2.44326236908334759821003489889, −1.61196525242618231958394644837,
1.98218441257278157562948435144, 2.81982988161587870516636917391, 4.63503847997264811832697588540, 5.82023031197378477795229859985, 6.58813910455144752810993759718, 8.035887433660023830391038933556, 8.499840279017852408162036496311, 9.124190845870337530166359597231, 9.902668479959372306671139606869, 10.68981127632341168138806434094