L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−2 − 3.46i)5-s + (1 − 1.73i)7-s − 0.999·8-s − 3.99·10-s + (0.5 − 0.866i)11-s + (−2 − 3.46i)13-s + (−0.999 − 1.73i)14-s + (−0.5 + 0.866i)16-s + 2·17-s + (−1.99 + 3.46i)20-s + (−0.499 − 0.866i)22-s + (−3 − 5.19i)23-s + (−5.49 + 9.52i)25-s − 3.99·26-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.894 − 1.54i)5-s + (0.377 − 0.654i)7-s − 0.353·8-s − 1.26·10-s + (0.150 − 0.261i)11-s + (−0.554 − 0.960i)13-s + (−0.267 − 0.462i)14-s + (−0.125 + 0.216i)16-s + 0.485·17-s + (−0.447 + 0.774i)20-s + (−0.106 − 0.184i)22-s + (−0.625 − 1.08i)23-s + (−1.09 + 1.90i)25-s − 0.784·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1782 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1782 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.253706140\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.253706140\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (2 + 3.46i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-1 + 1.73i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (2 + 3.46i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-5 + 8.66i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + (-1 - 1.73i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1 - 1.73i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6 - 10.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + (5 - 8.66i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2 + 3.46i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + (-1 + 1.73i)T + (-48.5 - 84.0i)T^{2} \) |
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
−8.583637301584287358600173477574, −8.236428942095635960785535768632, −7.48516776111054194574981804541, −6.20124048936886692863174149386, −5.15963777376743752362821344096, −4.57812557398747709938080558119, −3.94668630082327638756335245756, −2.82080509885026622399796887698, −1.26763355310766944663153741260, −0.45187735916321637433160989982,
2.11279806475012210660173338725, 3.13926476792238095901707336724, 3.93418526743128107267787605499, 4.86733137160445868796496096732, 5.89339241672310211268158905457, 6.73886920375161068278104658448, 7.27918130825428895188053975787, 7.935888989465890471542282661003, 8.786733691066238009365625646155, 9.768164956688600373656136190503