L(s) = 1 | + (0.5 + 0.866i)2-s + (0.500 − 0.866i)4-s + (−0.5 + 0.866i)5-s + 3·8-s − 0.999·10-s + (2.5 + 4.33i)11-s + (2.5 − 4.33i)13-s + (0.500 + 0.866i)16-s − 3·17-s + 19-s + (0.499 + 0.866i)20-s + (−2.5 + 4.33i)22-s + (1.5 − 2.59i)23-s + (2 + 3.46i)25-s + 5·26-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.250 − 0.433i)4-s + (−0.223 + 0.387i)5-s + 1.06·8-s − 0.316·10-s + (0.753 + 1.30i)11-s + (0.693 − 1.20i)13-s + (0.125 + 0.216i)16-s − 0.727·17-s + 0.229·19-s + (0.111 + 0.193i)20-s + (−0.533 + 0.923i)22-s + (0.312 − 0.541i)23-s + (0.400 + 0.692i)25-s + 0.980·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.400454052\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.400454052\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 + (2.5 - 4.33i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 9T + 53T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 3T + 73T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 13T + 89T^{2} \) |
| 97 | \( 1 + (-4.5 - 7.79i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.827786587342908299610877344975, −8.857317096816128570703841642266, −7.83496120326585387631539143519, −7.07778076857527006054714210820, −6.53391051084245554000618588398, −5.61560984657135122140564774575, −4.74613736808218236622065223945, −3.85972035716444058078916352829, −2.53259380379067120516362596354, −1.23422931124403506956313915831,
1.13297985530683275244795338550, 2.34070684351117564245518225379, 3.59274358122106589588075045065, 4.04459688183498998307446129871, 5.12198921335811558057689442127, 6.36265458883412573163699775731, 6.94763899406371646143018562965, 8.123691234363019259183943381850, 8.716782766916085427578498064222, 9.425299576344295973716770389234