L(s) = 1 | + (−0.373 + 1.36i)2-s + (−1.72 − 1.02i)4-s + (−1.28 − 0.743i)5-s + (1.36 − 2.26i)7-s + (2.03 − 1.96i)8-s + (1.49 − 1.47i)10-s + (1.37 + 2.37i)11-s − 5.10·13-s + (2.57 + 2.71i)14-s + (1.91 + 3.50i)16-s + (−4.52 + 2.61i)17-s + (−1.30 − 0.754i)19-s + (1.45 + 2.59i)20-s + (−3.75 + 0.983i)22-s + (−4.49 + 7.77i)23-s + ⋯ |
L(s) = 1 | + (−0.264 + 0.964i)2-s + (−0.860 − 0.510i)4-s + (−0.575 − 0.332i)5-s + (0.517 − 0.855i)7-s + (0.719 − 0.694i)8-s + (0.472 − 0.467i)10-s + (0.413 + 0.716i)11-s − 1.41·13-s + (0.688 + 0.725i)14-s + (0.479 + 0.877i)16-s + (−1.09 + 0.634i)17-s + (−0.299 − 0.172i)19-s + (0.325 + 0.579i)20-s + (−0.800 + 0.209i)22-s + (−0.936 + 1.62i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 + 0.526i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.849 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0280020 - 0.0983252i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0280020 - 0.0983252i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.373 - 1.36i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.36 + 2.26i)T \) |
good | 5 | \( 1 + (1.28 + 0.743i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.37 - 2.37i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 5.10T + 13T^{2} \) |
| 17 | \( 1 + (4.52 - 2.61i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.30 + 0.754i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.49 - 7.77i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2.11iT - 29T^{2} \) |
| 31 | \( 1 + (0.202 - 0.117i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.50 - 4.34i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 7.96iT - 41T^{2} \) |
| 43 | \( 1 + 6.52iT - 43T^{2} \) |
| 47 | \( 1 + (2.12 - 3.68i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.44 + 1.98i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.339 - 0.587i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.29 - 9.17i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (9.34 - 5.39i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 1.32T + 71T^{2} \) |
| 73 | \( 1 + (1.75 + 3.03i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-14.4 - 8.35i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 + (10.8 + 6.27i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−10.55434056905525426103287307311, −9.896782291546887721732789090195, −8.982073836303313890898329898031, −8.096941391489962867302651653912, −7.35819660304083469191892902382, −6.82403414691300332953953893649, −5.49626662522766805682350546009, −4.49402087787293981513977751924, −4.00309318544950397201921454853, −1.75784322604802365166893564668,
0.05525098143791211549788919300, 2.08155071641374483961266260777, 2.91988395751645210464885227139, 4.23052542799689367911446304710, 4.97802909677685048130714289997, 6.30516387825393303125144898797, 7.54767026862878574615629958835, 8.312162917682732800082468129132, 9.061260673942454352763233615124, 9.864153542647171134118434875640