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 + (3 + 5.19i)11-s + (1 − 1.73i)13-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s − 4·19-s + (3 − 5.19i)22-s + (−4.5 + 7.79i)23-s − 1.99·26-s + 0.999·28-s + (1.5 + 2.59i)29-s + (2 − 3.46i)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 + (0.904 + 1.56i)11-s + (0.277 − 0.480i)13-s + (−0.133 + 0.231i)14-s + (−0.125 − 0.216i)16-s − 0.917·19-s + (0.639 − 1.10i)22-s + (−0.938 + 1.62i)23-s − 0.392·26-s + 0.188·28-s + (0.278 + 0.482i)29-s + (0.359 − 0.622i)31-s + (−0.0883 + 0.153i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.098505726\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.098505726\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 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 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.5 - 2.59i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.5 - 11.2i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.859174724820605227048372779751, −9.065483656997210116745164352971, −8.171852383028030900330102108034, −7.31090421310699097285698098629, −6.61390421211242479918307628743, −5.44213448760577915414703800134, −4.27563843815304877202730897040, −3.70091175476191681069026203336, −2.32873392221335366537362804982, −1.31673424141656041364256313145,
0.54995543056266797734947930049, 2.11515490376249954656784950438, 3.53138756085233437597713095755, 4.41406478238854838983456340249, 5.64246077557228315658256775443, 6.33679087051504731293784758384, 6.83451291510766348085593582433, 8.172200978028912142730991282443, 8.673643113194546238716509087692, 9.164333772642387192789771684480