L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)7-s − 0.999·8-s + (−2.5 − 4.33i)13-s + (0.499 + 0.866i)14-s + (−0.5 + 0.866i)16-s − 3·17-s + 2·19-s + (−4.5 − 7.79i)23-s + (2.5 − 4.33i)25-s − 5·26-s + 0.999·28-s + (−1.5 + 2.59i)29-s + (−2.5 − 4.33i)31-s + (0.499 + 0.866i)32-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.188 + 0.327i)7-s − 0.353·8-s + (−0.693 − 1.20i)13-s + (0.133 + 0.231i)14-s + (−0.125 + 0.216i)16-s − 0.727·17-s + 0.458·19-s + (−0.938 − 1.62i)23-s + (0.5 − 0.866i)25-s − 0.980·26-s + 0.188·28-s + (−0.278 + 0.482i)29-s + (−0.449 − 0.777i)31-s + (0.0883 + 0.153i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.081892474\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.081892474\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + (4.5 + 7.79i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.5 - 11.2i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 + (4 - 6.92i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.668560844692927382484509656630, −8.696429661124892141416874886261, −7.951236135915009545633414434809, −6.81751324330959583993421152007, −5.92166960088236342353435483151, −5.04999033406061977695829306295, −4.15878802283952430183615936459, −2.97517000591496477531489828277, −2.18334898427611332614545841639, −0.39941090450697708471924845253,
1.79398034052069660191843039230, 3.25039431342898257412006554155, 4.19752837815257050096715567336, 5.05883848137187692748835115015, 6.00348404784966079694422444278, 6.98406452730295600993489160984, 7.41594919424181699244440397273, 8.477560232599063629792787379997, 9.391262284233501507210926728986, 9.885134133042836330935516795291