L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + 3.46i·5-s + (0.866 − 0.5i)7-s + 0.999i·8-s + (−1.73 + 2.99i)10-s + (3.46 + 2i)11-s + (−2.59 + 2.5i)13-s + 0.999·14-s + (−0.5 + 0.866i)16-s + (0.232 + 0.401i)17-s + (−0.464 + 0.267i)19-s + (−2.99 + 1.73i)20-s + (1.99 + 3.46i)22-s + (−1.13 + 1.96i)23-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + 1.54i·5-s + (0.327 − 0.188i)7-s + 0.353i·8-s + (−0.547 + 0.948i)10-s + (1.04 + 0.603i)11-s + (−0.720 + 0.693i)13-s + 0.267·14-s + (−0.125 + 0.216i)16-s + (0.0562 + 0.0974i)17-s + (−0.106 + 0.0614i)19-s + (−0.670 + 0.387i)20-s + (0.426 + 0.738i)22-s + (−0.236 + 0.409i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 - 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.398916938\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.398916938\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.866 + 0.5i)T \) |
| 13 | \( 1 + (2.59 - 2.5i)T \) |
good | 5 | \( 1 - 3.46iT - 5T^{2} \) |
| 11 | \( 1 + (-3.46 - 2i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.232 - 0.401i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.464 - 0.267i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.13 - 1.96i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4 + 6.92i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 0.464iT - 31T^{2} \) |
| 37 | \( 1 + (7.73 + 4.46i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-6 - 3.46i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.23 + 2.13i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 7.46iT - 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 + (8.42 - 4.86i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.59 + 4.5i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.76 - 1.59i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (10.3 - 5.96i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 2.92iT - 73T^{2} \) |
| 79 | \( 1 + 2.53T + 79T^{2} \) |
| 83 | \( 1 + 1.73iT - 83T^{2} \) |
| 89 | \( 1 + (7.16 + 4.13i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.26 + 4.19i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.807535935344387396369437901338, −8.870534505732894952614931579635, −7.65938833161333710079055272769, −7.18608992486505705371365049580, −6.52721481268586565573078432019, −5.81701853173329280080736364529, −4.51133628571109532771582843935, −3.92076257294677832733492709618, −2.83009841478646071295140190427, −1.90353373327086954508893271009,
0.75656266121290269311934935199, 1.75941581814683348079673352987, 3.12162205656384707986141094911, 4.16962241244482876833634540900, 4.95167586985191917313379168070, 5.49116764581241751495005278717, 6.46716936063112618777092756207, 7.52015229786117856521398375895, 8.677567296440278019246218820633, 8.820575980460263507306200426881