L(s) = 1 | + i·2-s + (2.75 + 1.59i)3-s − 4-s + (−2.23 + 0.0559i)5-s + (−1.59 + 2.75i)6-s + (−3.76 + 2.17i)7-s − i·8-s + (3.57 + 6.18i)9-s + (−0.0559 − 2.23i)10-s + 3.05·11-s + (−2.75 − 1.59i)12-s + (2.18 − 1.25i)13-s + (−2.17 − 3.76i)14-s + (−6.25 − 3.40i)15-s + 16-s + (−4.35 + 2.51i)17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (1.59 + 0.919i)3-s − 0.5·4-s + (−0.999 + 0.0250i)5-s + (−0.650 + 1.12i)6-s + (−1.42 + 0.822i)7-s − 0.353i·8-s + (1.19 + 2.06i)9-s + (−0.0176 − 0.706i)10-s + 0.921·11-s + (−0.796 − 0.459i)12-s + (0.604 − 0.349i)13-s + (−0.581 − 1.00i)14-s + (−1.61 − 0.879i)15-s + 0.250·16-s + (−1.05 + 0.609i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 - 0.476i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.878 - 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.396020 + 1.56053i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.396020 + 1.56053i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 + (2.23 - 0.0559i)T \) |
| 43 | \( 1 + (4.82 - 4.44i)T \) |
good | 3 | \( 1 + (-2.75 - 1.59i)T + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (3.76 - 2.17i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 - 3.05T + 11T^{2} \) |
| 13 | \( 1 + (-2.18 + 1.25i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (4.35 - 2.51i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.128 - 0.222i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.01 - 0.585i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.86 + 3.23i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.725 + 1.25i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.24 - 1.87i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 12.5T + 41T^{2} \) |
| 47 | \( 1 + 3.16iT - 47T^{2} \) |
| 53 | \( 1 + (-6.49 - 3.74i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 4.97T + 59T^{2} \) |
| 61 | \( 1 + (-5.56 - 9.63i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.17 - 4.72i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (7.80 + 13.5i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.16 + 1.24i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.92 - 6.79i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (12.5 + 7.22i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.69 + 6.39i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 18.7iT - 97T^{2} \) |
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
−11.46455226249157776340439874333, −10.24301796406322258296728900241, −9.329531993718462871071111440558, −8.832605378629165845334039730214, −8.163070583535539029534219632463, −7.06242639154579256931038602359, −5.99321729766350533787794629587, −4.32194049107914411678866972150, −3.72764015622164936700320284674, −2.74093294441097336338693372669,
0.908627070334034118798993297365, 2.59114373664757504149999194877, 3.64279078182402047161657273102, 4.07053536225436880268299496847, 6.66958504493311580890551433616, 7.04773272838792336084246575922, 8.181675484633732169992533409391, 9.078560750157439677346961693270, 9.507010651967265555947949244932, 10.85648169822536727451729554171