L(s) = 1 | + (0.821 − 1.52i)3-s + (−2.16 − 1.24i)5-s − 3.30·7-s + (−1.64 − 2.50i)9-s + 5.60i·11-s + (0.0750 − 0.0433i)13-s + (−3.67 + 2.26i)15-s + (4.69 + 2.71i)17-s + (−2.61 − 3.48i)19-s + (−2.71 + 5.04i)21-s + (−2.94 + 1.70i)23-s + (0.612 + 1.06i)25-s + (−5.17 + 0.453i)27-s + (1.78 + 3.09i)29-s + 6.88i·31-s + ⋯ |
L(s) = 1 | + (0.474 − 0.880i)3-s + (−0.966 − 0.557i)5-s − 1.25·7-s + (−0.549 − 0.835i)9-s + 1.69i·11-s + (0.0208 − 0.0120i)13-s + (−0.949 + 0.585i)15-s + (1.13 + 0.657i)17-s + (−0.600 − 0.799i)19-s + (−0.593 + 1.10i)21-s + (−0.614 + 0.354i)23-s + (0.122 + 0.212i)25-s + (−0.996 + 0.0872i)27-s + (0.332 + 0.575i)29-s + 1.23i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.245 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.169432 + 0.217664i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.169432 + 0.217664i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.821 + 1.52i)T \) |
| 19 | \( 1 + (2.61 + 3.48i)T \) |
good | 5 | \( 1 + (2.16 + 1.24i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + 3.30T + 7T^{2} \) |
| 11 | \( 1 - 5.60iT - 11T^{2} \) |
| 13 | \( 1 + (-0.0750 + 0.0433i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-4.69 - 2.71i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (2.94 - 1.70i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.78 - 3.09i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.88iT - 31T^{2} \) |
| 37 | \( 1 + 1.83iT - 37T^{2} \) |
| 41 | \( 1 + (1.35 - 2.33i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.02 - 6.97i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (8.16 - 4.71i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.741 + 1.28i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.91 + 11.9i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.28 + 10.8i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.13 + 4.11i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.20 - 9.02i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (4.19 - 7.26i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (11.1 + 6.42i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 11.1iT - 83T^{2} \) |
| 89 | \( 1 + (0.0767 + 0.132i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.41 - 1.39i)T + (48.5 + 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.02887827673366888362333626502, −9.524855628969761391804086169879, −8.463643887146919610829203279809, −7.84385770866229167473356181024, −6.95850699128733218575443078784, −6.38805732147201590716333953847, −4.97868156012460412562146329802, −3.87473920631839030125720805151, −2.98191551200899299662179121975, −1.56185139917476233542912499760,
0.12131512485039009937992629509, 2.79357502846393320223146999258, 3.47478875712466282957856026101, 4.03957903631941181882614687987, 5.56936345237386867892690985775, 6.28156908991025540646380429352, 7.51443308033205894985981412813, 8.243289664011277945325121823781, 8.987770546085434397819658137894, 10.06096343403088477293333379071