L(s) = 1 | + (0.766 + 0.642i)2-s − 3-s + (0.173 + 0.984i)4-s + (−0.766 − 0.642i)6-s + (−0.500 + 0.866i)8-s + 9-s + 0.684i·11-s + (−0.173 − 0.984i)12-s + (−0.939 + 0.342i)16-s + (0.592 + 1.62i)17-s + (0.766 + 0.642i)18-s + (−0.173 − 0.984i)19-s + (−0.439 + 0.524i)22-s + (0.500 − 0.866i)24-s + (−0.766 + 0.642i)25-s + ⋯ |
L(s) = 1 | + (0.766 + 0.642i)2-s − 3-s + (0.173 + 0.984i)4-s + (−0.766 − 0.642i)6-s + (−0.500 + 0.866i)8-s + 9-s + 0.684i·11-s + (−0.173 − 0.984i)12-s + (−0.939 + 0.342i)16-s + (0.592 + 1.62i)17-s + (0.766 + 0.642i)18-s + (−0.173 − 0.984i)19-s + (−0.439 + 0.524i)22-s + (0.500 − 0.866i)24-s + (−0.766 + 0.642i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.123642302\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.123642302\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 + (0.173 + 0.984i)T \) |
good | 5 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 - 0.684iT - T^{2} \) |
| 13 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 17 | \( 1 + (-0.592 - 1.62i)T + (-0.766 + 0.642i)T^{2} \) |
| 23 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 29 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-1.43 - 1.20i)T + (0.173 + 0.984i)T^{2} \) |
| 43 | \( 1 + (0.347 - 1.96i)T + (-0.939 - 0.342i)T^{2} \) |
| 47 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 53 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 59 | \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \) |
| 61 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 67 | \( 1 + (0.439 + 0.524i)T + (-0.173 + 0.984i)T^{2} \) |
| 71 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 73 | \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \) |
| 79 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 83 | \( 1 + (-1.11 - 0.642i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 97 | \( 1 + (-1.26 + 1.50i)T + (-0.173 - 0.984i)T^{2} \) |
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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
−10.10076185682698760730284311655, −9.276865428825989572970320387813, −8.050855926725205333715794144918, −7.48928603042435754971644403814, −6.45582348949527323806379725548, −6.05419141831477764790828635024, −5.01612820964323366732857670994, −4.40035414254012509797823602235, −3.39449646324545225796979668459, −1.84054963731599005273577911957,
0.862286479228001711428707104195, 2.30881956085845287863328944923, 3.57964509186964109195549706119, 4.40770287557479677120210061926, 5.47011449175016314668827736227, 5.81689593578337409144251652239, 6.83849645864095500132667735449, 7.65893978601297713622308396450, 9.019025428462849275155805269778, 9.872753009095947601101015777709