L(s) = 1 | + (−0.207 − 0.359i)2-s + (0.913 − 1.58i)4-s + (1.10 − 1.91i)5-s + (0.659 + 1.14i)7-s − 1.59·8-s − 0.920·10-s + (−2.60 − 4.51i)11-s + (0.00902 − 0.0156i)13-s + (0.274 − 0.474i)14-s + (−1.49 − 2.59i)16-s + 3.13·17-s + 0.417·19-s + (−2.02 − 3.50i)20-s + (−1.08 + 1.87i)22-s + (−0.517 + 0.895i)23-s + ⋯ |
L(s) = 1 | + (−0.146 − 0.254i)2-s + (0.456 − 0.791i)4-s + (0.495 − 0.857i)5-s + (0.249 + 0.431i)7-s − 0.562·8-s − 0.291·10-s + (−0.786 − 1.36i)11-s + (0.00250 − 0.00433i)13-s + (0.0732 − 0.126i)14-s + (−0.374 − 0.648i)16-s + 0.759·17-s + 0.0957·19-s + (−0.452 − 0.783i)20-s + (−0.230 + 0.400i)22-s + (−0.107 + 0.186i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.765752 - 1.32632i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.765752 - 1.32632i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + (0.207 + 0.359i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.10 + 1.91i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.659 - 1.14i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.60 + 4.51i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.00902 + 0.0156i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 3.13T + 17T^{2} \) |
| 19 | \( 1 - 0.417T + 19T^{2} \) |
| 23 | \( 1 + (0.517 - 0.895i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.90 + 6.76i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.86 - 3.22i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4.42T + 37T^{2} \) |
| 41 | \( 1 + (1.83 - 3.18i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.15 - 7.19i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.54 + 6.14i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 1.30T + 53T^{2} \) |
| 59 | \( 1 + (1.85 - 3.20i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.45 + 5.98i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.51 + 9.54i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6.08T + 71T^{2} \) |
| 73 | \( 1 + 0.546T + 73T^{2} \) |
| 79 | \( 1 + (0.244 + 0.423i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.30 - 3.99i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 3.37T + 89T^{2} \) |
| 97 | \( 1 + (4.97 + 8.60i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.03591865452686531535208303120, −9.372226550861473364355601563529, −8.527826892028473687861122579604, −7.68408070557342325216957612956, −6.23120845690974327341577306335, −5.61539572152523417338614023919, −4.98248417187141213418906902750, −3.26711684272036390865206722349, −2.06440481441846815997981337225, −0.818556591338197625627502081573,
2.03898575322194421021117425103, 2.97784538523869449850642382967, 4.15195635158513195590779009062, 5.42813255683226241996384009170, 6.52046743509210277377643861761, 7.39084146263659909615802684278, 7.68177851706213392768849558293, 8.938407395032632239306444058535, 9.961603769294057258363541105338, 10.59890005749010599770400801293