L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−1 − 1.73i)7-s + 0.999·8-s + (−0.5 − 0.866i)11-s + (2 − 3.46i)13-s + (−0.999 + 1.73i)14-s + (−0.5 − 0.866i)16-s + 6·17-s − 4·19-s + (−0.499 + 0.866i)22-s + (3 − 5.19i)23-s + (2.5 + 4.33i)25-s − 3.99·26-s + 1.99·28-s + (3 + 5.19i)29-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.377 − 0.654i)7-s + 0.353·8-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 + 1.45·17-s − 0.917·19-s + (−0.106 + 0.184i)22-s + (0.625 − 1.08i)23-s + (0.5 + 0.866i)25-s − 0.784·26-s + 0.377·28-s + (0.557 + 0.964i)29-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
\(L(1)\) |
\(\approx\) |
\(1.042346784\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.042346784\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 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 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + (7 + 12.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.738998430600416388610153338538, −8.581147888408567338006917788329, −7.39110199554316148995016744132, −6.82542037600547961452878804707, −5.62646410779144380575452825139, −4.84534639512729071827604618803, −3.44188592026593616479593879872, −3.25022639125648516736604416398, −1.63954905812993168332644798239, −0.47462220248973644891639370367,
1.35642919363662483937454838769, 2.63450042991239529065225519724, 3.84188982174272294829702736827, 4.81854688023509202976834512079, 5.81663100136610043237960515055, 6.33460235521880989030814341762, 7.24650665002079577333208976751, 8.055551709269473223421469210565, 8.765688808031857513565172206587, 9.520630331561239618151359779381