L(s) = 1 | + (−0.684 + 1.18i)3-s + (0.780 − 1.35i)5-s + (0.561 + 0.972i)9-s − 3.50·11-s + (−0.219 − 0.379i)13-s + (1.06 + 1.85i)15-s + (0.219 − 0.379i)17-s + (−3.12 + 3.03i)19-s + (2.43 + 4.22i)23-s + (1.28 + 2.21i)25-s − 5.64·27-s + (2.34 + 4.05i)29-s + 2.73·31-s + (2.40 − 4.16i)33-s + 1.12·37-s + ⋯ |
L(s) = 1 | + (−0.395 + 0.684i)3-s + (0.349 − 0.604i)5-s + (0.187 + 0.324i)9-s − 1.05·11-s + (−0.0608 − 0.105i)13-s + (0.276 + 0.478i)15-s + (0.0531 − 0.0920i)17-s + (−0.716 + 0.697i)19-s + (0.508 + 0.881i)23-s + (0.256 + 0.443i)25-s − 1.08·27-s + (0.434 + 0.753i)29-s + 0.492·31-s + (0.418 − 0.724i)33-s + 0.184·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.038436395\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.038436395\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (3.12 - 3.03i)T \) |
good | 3 | \( 1 + (0.684 - 1.18i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.780 + 1.35i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 3.50T + 11T^{2} \) |
| 13 | \( 1 + (0.219 + 0.379i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.219 + 0.379i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-2.43 - 4.22i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.34 - 4.05i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 2.73T + 31T^{2} \) |
| 37 | \( 1 - 1.12T + 37T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.80 - 6.59i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.06 - 1.85i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.219 - 0.379i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.684 + 1.18i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.21 + 2.11i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.93 - 12.0i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.43 - 4.22i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.623 - 1.07i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.06 + 1.85i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 6.24T + 83T^{2} \) |
| 89 | \( 1 + (-5.34 - 9.25i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.62 - 9.73i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.994925995849952049432431944428, −9.388591931228191845755850688731, −8.351815449465270036728606736982, −7.70266043540801210118422425125, −6.57769279753598816988349611444, −5.39983451951945533811726287281, −5.10103508421786210570398589015, −4.11166258865235076864890157345, −2.86199543812646480683781857797, −1.49161751235416593924183436488,
0.46435937035163675947438230143, 2.08264371981845004653990417402, 2.96315952666570897498502079853, 4.34881978515194701260541971975, 5.34429625952117872511036583141, 6.45141197077749392455772186156, 6.71061306237431680381966883149, 7.73820114006778916792046388884, 8.536968828504466334834349700124, 9.578931395764229975034379153810