L(s) = 1 | − 2.64·7-s + 0.913i·11-s + 9.39i·23-s − 5·25-s + 6.06i·29-s − 10.5·37-s − 5.29·43-s + 7.00·49-s + 14.5i·53-s + 4·67-s − 7.57i·71-s − 2.41i·77-s − 8·79-s + 17.8i·107-s − 10.5·109-s + ⋯ |
L(s) = 1 | − 0.999·7-s + 0.275i·11-s + 1.95i·23-s − 25-s + 1.12i·29-s − 1.73·37-s − 0.806·43-s + 49-s + 1.99i·53-s + 0.488·67-s − 0.898i·71-s − 0.275i·77-s − 0.900·79-s + 1.72i·107-s − 1.01·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6914546887\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6914546887\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 0.913iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 9.39iT - 23T^{2} \) |
| 29 | \( 1 - 6.06iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 5.29T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 14.5iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 7.57iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−10.12136802681315335069849460796, −9.480367949428410688175354626745, −8.753358876457901673083993034534, −7.59318580227557593252802925469, −6.96426919512390221610445399513, −5.96839169623073884813748279640, −5.17105510266094715806725652569, −3.85898510986526687257939837318, −3.10401688515753659614802474690, −1.65201866572122924480582025909,
0.30318841008827177750682778802, 2.19213825274132137999363017202, 3.31313034992781075852075463251, 4.25130941245083514008844370268, 5.43652659758881910762848195399, 6.36722648633120806220838381120, 6.96488620287127490188964251850, 8.154485569345957292329858858796, 8.795884154079218316170134677675, 9.854435759971867152852786801040