L(s) = 1 | − 0.732i·7-s − 1.73·11-s − 1.46i·13-s + 1.26i·17-s − 2.46·19-s − 3.46i·23-s + 4.26·29-s − 7.92·31-s − 4.19i·37-s − 0.803·41-s + 6.73i·43-s + 4.73i·47-s + 6.46·49-s − 10.7i·53-s + 4.26·59-s + ⋯ |
L(s) = 1 | − 0.276i·7-s − 0.522·11-s − 0.406i·13-s + 0.307i·17-s − 0.565·19-s − 0.722i·23-s + 0.792·29-s − 1.42·31-s − 0.689i·37-s − 0.125·41-s + 1.02i·43-s + 0.690i·47-s + 0.923·49-s − 1.47i·53-s + 0.555·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6631598649\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6631598649\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 0.732iT - 7T^{2} \) |
| 11 | \( 1 + 1.73T + 11T^{2} \) |
| 13 | \( 1 + 1.46iT - 13T^{2} \) |
| 17 | \( 1 - 1.26iT - 17T^{2} \) |
| 19 | \( 1 + 2.46T + 19T^{2} \) |
| 23 | \( 1 + 3.46iT - 23T^{2} \) |
| 29 | \( 1 - 4.26T + 29T^{2} \) |
| 31 | \( 1 + 7.92T + 31T^{2} \) |
| 37 | \( 1 + 4.19iT - 37T^{2} \) |
| 41 | \( 1 + 0.803T + 41T^{2} \) |
| 43 | \( 1 - 6.73iT - 43T^{2} \) |
| 47 | \( 1 - 4.73iT - 47T^{2} \) |
| 53 | \( 1 + 10.7iT - 53T^{2} \) |
| 59 | \( 1 - 4.26T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 14.3iT - 67T^{2} \) |
| 71 | \( 1 + 0.803T + 71T^{2} \) |
| 73 | \( 1 - 10.1iT - 73T^{2} \) |
| 79 | \( 1 + 6.39T + 79T^{2} \) |
| 83 | \( 1 - 9.12iT - 83T^{2} \) |
| 89 | \( 1 - 5.19T + 89T^{2} \) |
| 97 | \( 1 - 2.73iT - 97T^{2} \) |
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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
−8.191418364033702981231771655848, −7.28541555526326933932916365389, −6.77962390103852327308084795634, −5.89939241315025358891396427610, −5.34200831167183200775491970569, −4.46196885201826263911358879286, −3.83911474139558682258485030435, −2.88771833616276332180968625895, −2.15401280280324063304424457901, −1.00600560768580006345935959345,
0.16648636980915045512913289406, 1.54321194333131325202879722906, 2.36772493154904317298346078560, 3.21989197836364247643755961681, 4.03687227080622406513105036695, 4.85672726409979976307751495546, 5.49831316211060559297721960132, 6.19261785736759631743087675243, 6.98379194946591855567571089124, 7.55656556184489410611633581574