L(s) = 1 | + 3-s + 5-s + (−1.09 − 1.89i)7-s + 9-s + (2.56 + 4.44i)11-s + (3.21 − 5.57i)13-s + 15-s + (0.653 − 1.13i)17-s + (−2.76 + 4.78i)19-s + (−1.09 − 1.89i)21-s + (−1.52 + 2.64i)23-s + 25-s + 27-s + (3.69 + 6.39i)29-s + (1.00 + 1.73i)31-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + (−0.413 − 0.716i)7-s + 0.333·9-s + (0.773 + 1.33i)11-s + (0.892 − 1.54i)13-s + 0.258·15-s + (0.158 − 0.274i)17-s + (−0.634 + 1.09i)19-s + (−0.238 − 0.413i)21-s + (−0.318 + 0.552i)23-s + 0.200·25-s + 0.192·27-s + (0.685 + 1.18i)29-s + (0.179 + 0.311i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.799197947\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.799197947\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (7.43 + 3.42i)T \) |
good | 7 | \( 1 + (1.09 + 1.89i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.56 - 4.44i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.21 + 5.57i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.653 + 1.13i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.76 - 4.78i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.52 - 2.64i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.69 - 6.39i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.00 - 1.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.40 + 2.44i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.19 - 2.07i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 - 9.64T + 43T^{2} \) |
| 47 | \( 1 + (-1.71 - 2.96i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 6.09T + 53T^{2} \) |
| 59 | \( 1 + 5.99T + 59T^{2} \) |
| 61 | \( 1 + (-6.06 + 10.5i)T + (-30.5 - 52.8i)T^{2} \) |
| 71 | \( 1 + (-7.59 - 13.1i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.15 + 10.6i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.94 - 8.55i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.69 + 11.5i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 5.88T + 89T^{2} \) |
| 97 | \( 1 + (-7.09 + 12.2i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.328069439391671297235933508853, −7.78239012261943290581801143991, −7.01185879468813889435879520346, −6.32940865372221710524243067997, −5.53317942723806535473456283179, −4.51834588153203844404604830940, −3.74445125302212516687158822163, −3.07882204818339752375132563517, −1.90416351144252133116365199145, −1.01918327572288818694718984856,
0.942756794459645541894288790746, 2.14485223228297874144035102496, 2.82550201279448741728862516732, 3.88576271497291049322820145688, 4.42662766369728256979837754281, 5.72857775842007750439779068100, 6.36970529150785924930873560371, 6.65057417722436173682002865769, 7.938130170849205876272352386431, 8.668404358101265381412118439652