L(s) = 1 | − 4.99i·3-s − 11.3·7-s − 15.9·9-s + 13.5i·11-s + 0.886i·13-s + 12.3i·17-s − 37.2·19-s + 56.7i·21-s + 24.9i·23-s + 34.5i·27-s − 24.1i·29-s + (−6.84 − 30.2i)31-s + 67.4·33-s + 50.7i·37-s + 4.42·39-s + ⋯ |
L(s) = 1 | − 1.66i·3-s − 1.62·7-s − 1.76·9-s + 1.22i·11-s + 0.0681i·13-s + 0.724i·17-s − 1.96·19-s + 2.70i·21-s + 1.08i·23-s + 1.27i·27-s − 0.832i·29-s + (−0.220 − 0.975i)31-s + 2.04·33-s + 1.37i·37-s + 0.113·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 + 0.975i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8035348848\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8035348848\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 + (6.84 + 30.2i)T \) |
good | 3 | \( 1 + 4.99iT - 9T^{2} \) |
| 7 | \( 1 + 11.3T + 49T^{2} \) |
| 11 | \( 1 - 13.5iT - 121T^{2} \) |
| 13 | \( 1 - 0.886iT - 169T^{2} \) |
| 17 | \( 1 - 12.3iT - 289T^{2} \) |
| 19 | \( 1 + 37.2T + 361T^{2} \) |
| 23 | \( 1 - 24.9iT - 529T^{2} \) |
| 29 | \( 1 + 24.1iT - 841T^{2} \) |
| 37 | \( 1 - 50.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 32.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + 56.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 40.4T + 2.20e3T^{2} \) |
| 53 | \( 1 + 18.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 15.8T + 3.48e3T^{2} \) |
| 61 | \( 1 + 61.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 38.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 98.4T + 5.04e3T^{2} \) |
| 73 | \( 1 - 63.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 73.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 95.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 162. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 177.T + 9.40e3T^{2} \) |
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
−8.246429830547981390074091031384, −7.39875401743385979100324015416, −6.85525944945359037773944879194, −6.30121855216089902342998459977, −5.75904201242282176120295025882, −4.36233787805007327325513808351, −3.43869264071391568908378060172, −2.32019978882781511291571567243, −1.80643057973286264190423323044, −0.39215686366877123751209081628,
0.43262869483434382382259770655, 2.60150891294061656190570453671, 3.24284372737862113664599706570, 3.91536742974852754238263804789, 4.67796085512161573511354182074, 5.63752773719930847729703953734, 6.25711772702179962770817499312, 6.96166986105980107617203598259, 8.342217881746901657533110170496, 8.981895574955191908769040900323