L(s) = 1 | + (0.430 − 1.34i)2-s − 2.96·3-s + (−1.62 − 1.15i)4-s + (−0.177 − 2.22i)5-s + (−1.27 + 3.99i)6-s + (−0.115 − 0.115i)7-s + (−2.26 + 1.69i)8-s + 5.79·9-s + (−3.07 − 0.720i)10-s + (2.95 − 2.95i)11-s + (4.83 + 3.43i)12-s − 1.55i·13-s + (−0.204 + 0.105i)14-s + (0.525 + 6.61i)15-s + (1.31 + 3.77i)16-s + (0.299 + 0.299i)17-s + ⋯ |
L(s) = 1 | + (0.304 − 0.952i)2-s − 1.71·3-s + (−0.814 − 0.579i)4-s + (−0.0793 − 0.996i)5-s + (−0.520 + 1.63i)6-s + (−0.0435 − 0.0435i)7-s + (−0.800 + 0.599i)8-s + 1.93·9-s + (−0.973 − 0.227i)10-s + (0.892 − 0.892i)11-s + (1.39 + 0.992i)12-s − 0.432i·13-s + (−0.0546 + 0.0282i)14-s + (0.135 + 1.70i)15-s + (0.327 + 0.944i)16-s + (0.0726 + 0.0726i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 + 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.164922 - 0.558417i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.164922 - 0.558417i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.430 + 1.34i)T \) |
| 5 | \( 1 + (0.177 + 2.22i)T \) |
good | 3 | \( 1 + 2.96T + 3T^{2} \) |
| 7 | \( 1 + (0.115 + 0.115i)T + 7iT^{2} \) |
| 11 | \( 1 + (-2.95 + 2.95i)T - 11iT^{2} \) |
| 13 | \( 1 + 1.55iT - 13T^{2} \) |
| 17 | \( 1 + (-0.299 - 0.299i)T + 17iT^{2} \) |
| 19 | \( 1 + (2.26 - 2.26i)T - 19iT^{2} \) |
| 23 | \( 1 + (-4.14 + 4.14i)T - 23iT^{2} \) |
| 29 | \( 1 + (-0.289 - 0.289i)T + 29iT^{2} \) |
| 31 | \( 1 + 4.18iT - 31T^{2} \) |
| 37 | \( 1 - 1.63iT - 37T^{2} \) |
| 41 | \( 1 - 7.61iT - 41T^{2} \) |
| 43 | \( 1 - 6.72iT - 43T^{2} \) |
| 47 | \( 1 + (-4.38 + 4.38i)T - 47iT^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 + (-1.63 - 1.63i)T + 59iT^{2} \) |
| 61 | \( 1 + (1.23 - 1.23i)T - 61iT^{2} \) |
| 67 | \( 1 - 2.49iT - 67T^{2} \) |
| 71 | \( 1 - 8.00T + 71T^{2} \) |
| 73 | \( 1 + (1.12 + 1.12i)T + 73iT^{2} \) |
| 79 | \( 1 - 3.62T + 79T^{2} \) |
| 83 | \( 1 - 1.62T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 + (-9.69 - 9.69i)T + 97iT^{2} \) |
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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
−13.39760190120175615566754698764, −12.56980963206724800310217852314, −11.77549741483412631279254026737, −11.00952335081074125359408640808, −9.911148282393493865064393454895, −8.566928556849731653148284917813, −6.28277968221160283579571724451, −5.30887662968168042618607220184, −4.14657684899582333219578443174, −0.916915229086749418346729581447,
4.15601253177353902389854066932, 5.49639045490722261691543526730, 6.70117402993890786866802353201, 7.17016833798032541184106816505, 9.328088808244823761777077118468, 10.62682574478976711029806455803, 11.72737978853206026884777954360, 12.52816476296601895860674385827, 13.91549983801695502107008539384, 15.08522907344100390543956213143