L(s) = 1 | + (−1.32 − 2.29i)2-s + (−2.5 + 4.33i)4-s + (1.32 − 2.29i)5-s + (0.5 + 0.866i)7-s + 7.93·8-s − 7·10-s + (1.32 + 2.29i)11-s + (1 − 1.73i)13-s + (1.32 − 2.29i)14-s + (−5.49 − 9.52i)16-s + 7·19-s + (6.61 + 11.4i)20-s + (3.5 − 6.06i)22-s + (3.96 − 6.87i)23-s + (−1 − 1.73i)25-s − 5.29·26-s + ⋯ |
L(s) = 1 | + (−0.935 − 1.62i)2-s + (−1.25 + 2.16i)4-s + (0.591 − 1.02i)5-s + (0.188 + 0.327i)7-s + 2.80·8-s − 2.21·10-s + (0.398 + 0.690i)11-s + (0.277 − 0.480i)13-s + (0.353 − 0.612i)14-s + (−1.37 − 2.38i)16-s + 1.60·19-s + (1.47 + 2.56i)20-s + (0.746 − 1.29i)22-s + (0.827 − 1.43i)23-s + (−0.200 − 0.346i)25-s − 1.03·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.336945 - 0.925749i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.336945 - 0.925749i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (1.32 + 2.29i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.32 + 2.29i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.32 - 2.29i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 + (-3.96 + 6.87i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.64 + 4.58i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 + (1.32 - 2.29i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 7.93T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (7.93 + 13.7i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 18.5T + 89T^{2} \) |
| 97 | \( 1 + (-6 - 10.3i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.25005659911440755842250918357, −9.620626531185860850837736262684, −8.915006541109687133947429287191, −8.322130705159685853565238178871, −7.17430625861009816682813996908, −5.42585368077153635625500422181, −4.49384331146462366318721638305, −3.20829445335908572224602172141, −1.97695338729716011605404330780, −0.940417049047063968904108612557,
1.34722206450387431720409302345, 3.44361010995422362270805818530, 5.14753551535851252444716110218, 5.88626963209246009090443632817, 6.84347978611362862299762395545, 7.29406964875944349046920551781, 8.302624355901766592492031115991, 9.335808294571976032115683915082, 9.764381609410089934901888233530, 10.83280917659457455559598685099