L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.5 − 2.59i)5-s + 0.999·8-s − 3·10-s + (−1.5 − 2.59i)11-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)16-s − 3·17-s − 7·19-s + (1.50 + 2.59i)20-s + (−1.5 + 2.59i)22-s + (−4.5 + 7.79i)23-s + (−2 − 3.46i)25-s − 0.999·26-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.670 − 1.16i)5-s + 0.353·8-s − 0.948·10-s + (−0.452 − 0.783i)11-s + (0.138 − 0.240i)13-s + (−0.125 − 0.216i)16-s − 0.727·17-s − 1.60·19-s + (0.335 + 0.580i)20-s + (−0.319 + 0.553i)22-s + (−0.938 + 1.62i)23-s + (−0.400 − 0.692i)25-s − 0.196·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + 7T + 19T^{2} \) |
| 23 | \( 1 + (4.5 - 7.79i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.525981555891394904671065376501, −7.914764383414332901513252107540, −6.77092911531093936737210179287, −5.80325274642630787284168306162, −5.18493081986776629969099621857, −4.27223890798163676500384633114, −3.34454478419446373355140836559, −2.15025316501747392214106086240, −1.35895503129630697372067802881, 0,
2.07654012532843986372758369815, 2.47663939708395164456329372230, 4.03727091623910549800775748103, 4.70481232263672350838600291987, 5.97247407287435424106332010856, 6.38488890370501899534161580181, 6.97984422771655409719216593360, 7.85093026122805071149489498165, 8.570193698778401875116366566303