L(s) = 1 | + (3.23 + 3.23i)7-s + 4.57i·11-s + (4.23 − 4.23i)13-s + (1.74 − 1.74i)17-s − 2.47i·19-s + (2.82 + 2.82i)23-s − 5.99·29-s + 1.52·31-s + (−2.23 − 2.23i)37-s + 7.07i·41-s + (−2.47 + 2.47i)43-s + (−1.74 + 1.74i)47-s + 13.9i·49-s − 1.08·59-s + 10.4·61-s + ⋯ |
L(s) = 1 | + (1.22 + 1.22i)7-s + 1.37i·11-s + (1.17 − 1.17i)13-s + (0.423 − 0.423i)17-s − 0.567i·19-s + (0.589 + 0.589i)23-s − 1.11·29-s + 0.274·31-s + (−0.367 − 0.367i)37-s + 1.10i·41-s + (−0.376 + 0.376i)43-s + (−0.254 + 0.254i)47-s + 1.99i·49-s − 0.140·59-s + 1.34·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.128743813\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.128743813\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-3.23 - 3.23i)T + 7iT^{2} \) |
| 11 | \( 1 - 4.57iT - 11T^{2} \) |
| 13 | \( 1 + (-4.23 + 4.23i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.74 + 1.74i)T - 17iT^{2} \) |
| 19 | \( 1 + 2.47iT - 19T^{2} \) |
| 23 | \( 1 + (-2.82 - 2.82i)T + 23iT^{2} \) |
| 29 | \( 1 + 5.99T + 29T^{2} \) |
| 31 | \( 1 - 1.52T + 31T^{2} \) |
| 37 | \( 1 + (2.23 + 2.23i)T + 37iT^{2} \) |
| 41 | \( 1 - 7.07iT - 41T^{2} \) |
| 43 | \( 1 + (2.47 - 2.47i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.74 - 1.74i)T - 47iT^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 + 1.08T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 + (-1.52 - 1.52i)T + 67iT^{2} \) |
| 71 | \( 1 + 12.6iT - 71T^{2} \) |
| 73 | \( 1 + (-0.527 + 0.527i)T - 73iT^{2} \) |
| 79 | \( 1 - 14.4iT - 79T^{2} \) |
| 83 | \( 1 + (-1.08 - 1.08i)T + 83iT^{2} \) |
| 89 | \( 1 + 0.746T + 89T^{2} \) |
| 97 | \( 1 + (1 + i)T + 97iT^{2} \) |
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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
−9.323353728206380936372257526540, −8.535589829680385746461767113990, −7.899069120030445941895833731835, −7.16198497749292804530885772416, −5.99095518811081890456706696636, −5.26695039914892397748431446104, −4.68745615023557036597946563595, −3.39015230605912291563701637769, −2.32457849100216075874705903593, −1.34135362452321793471181020997,
0.928181284542880839240623007285, 1.83439643981556491452387915965, 3.55161907898791844307188370511, 3.99970934117968764846979101785, 5.06212656635308692521089076547, 5.97728426204619667862197829477, 6.81983208422755050998220807464, 7.66630159726220621563619587074, 8.472572745349686984076339882553, 8.859126698391304609228821266390