L(s) = 1 | − 1.09i·2-s + (1.70 − 0.323i)3-s + 0.791·4-s + (−0.355 − 1.87i)6-s + (1 + 2.44i)7-s − 3.06i·8-s + (2.79 − 1.09i)9-s + 3.06i·11-s + (1.34 − 0.255i)12-s − 2.44i·13-s + (2.69 − 1.09i)14-s − 1.79·16-s − 2.69·17-s + (−1.20 − 3.06i)18-s + 4.38i·19-s + ⋯ |
L(s) = 1 | − 0.777i·2-s + (0.982 − 0.186i)3-s + 0.395·4-s + (−0.144 − 0.763i)6-s + (0.377 + 0.925i)7-s − 1.08i·8-s + (0.930 − 0.366i)9-s + 0.925i·11-s + (0.388 − 0.0737i)12-s − 0.679i·13-s + (0.719 − 0.293i)14-s − 0.447·16-s − 0.653·17-s + (−0.284 − 0.723i)18-s + 1.00i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.544 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.544 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.07785 - 1.12920i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.07785 - 1.12920i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.70 + 0.323i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1 - 2.44i)T \) |
good | 2 | \( 1 + 1.09iT - 2T^{2} \) |
| 11 | \( 1 - 3.06iT - 11T^{2} \) |
| 13 | \( 1 + 2.44iT - 13T^{2} \) |
| 17 | \( 1 + 2.69T + 17T^{2} \) |
| 19 | \( 1 - 4.38iT - 19T^{2} \) |
| 23 | \( 1 + 5.26iT - 23T^{2} \) |
| 29 | \( 1 + 5.26iT - 29T^{2} \) |
| 31 | \( 1 - 6.83iT - 31T^{2} \) |
| 37 | \( 1 + 8.58T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 - 6.58T + 43T^{2} \) |
| 47 | \( 1 + 2.69T + 47T^{2} \) |
| 53 | \( 1 - 3.93iT - 53T^{2} \) |
| 59 | \( 1 - 7.51T + 59T^{2} \) |
| 61 | \( 1 + 6.83iT - 61T^{2} \) |
| 67 | \( 1 + 4.16T + 67T^{2} \) |
| 71 | \( 1 - 3.06iT - 71T^{2} \) |
| 73 | \( 1 + 16.1iT - 73T^{2} \) |
| 79 | \( 1 - 0.582T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 + 7.51T + 89T^{2} \) |
| 97 | \( 1 - 11.7iT - 97T^{2} \) |
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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
−10.52705596669300936653233827567, −10.01952471990102179938953070907, −8.970001132797234004070895844727, −8.193691995956905966784282171187, −7.20794963473525024753526443403, −6.27046746319538881498878689837, −4.78048416539688629214989722120, −3.54205074566745684303065796964, −2.48218510391075962782045019821, −1.70946924004765435134188156464,
1.78151578642576374639183407985, 3.15803913504395878615806680332, 4.31508523882146213789451170327, 5.44578560018060438042243814153, 6.87888688228404847939200626128, 7.22228281580004200523058795502, 8.322742834739307581618142809863, 8.871443233604009003909222700206, 10.03597942986856504058147893817, 11.05341415430166242175512453242