| L(s) = 1 | + (1.16 + 0.795i)2-s + (−0.342 + 0.939i)3-s + (0.734 + 1.86i)4-s + (−0.853 + 0.150i)5-s + (−1.14 + 0.826i)6-s + (0.0398 − 0.0690i)7-s + (−0.621 + 2.75i)8-s + (−0.766 − 0.642i)9-s + (−1.11 − 0.503i)10-s + (−0.739 + 0.427i)11-s + (−1.99 + 0.0535i)12-s + (2.18 + 6.01i)13-s + (0.101 − 0.0489i)14-s + (0.150 − 0.853i)15-s + (−2.92 + 2.73i)16-s + (0.464 − 0.389i)17-s + ⋯ |
| L(s) = 1 | + (0.826 + 0.562i)2-s + (−0.197 + 0.542i)3-s + (0.367 + 0.930i)4-s + (−0.381 + 0.0673i)5-s + (−0.468 + 0.337i)6-s + (0.0150 − 0.0260i)7-s + (−0.219 + 0.975i)8-s + (−0.255 − 0.214i)9-s + (−0.353 − 0.159i)10-s + (−0.223 + 0.128i)11-s + (−0.577 + 0.0154i)12-s + (0.607 + 1.66i)13-s + (0.0271 − 0.0130i)14-s + (0.0388 − 0.220i)15-s + (−0.730 + 0.682i)16-s + (0.112 − 0.0944i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.726228 + 1.66196i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.726228 + 1.66196i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.16 - 0.795i)T \) |
| 3 | \( 1 + (0.342 - 0.939i)T \) |
| 19 | \( 1 + (3.00 - 3.15i)T \) |
| good | 5 | \( 1 + (0.853 - 0.150i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.0398 + 0.0690i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.739 - 0.427i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.18 - 6.01i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.464 + 0.389i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.655 + 3.71i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-4.15 + 4.94i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-5.02 + 8.69i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 0.193iT - 37T^{2} \) |
| 41 | \( 1 + (-5.97 - 2.17i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-3.40 + 0.600i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-2.84 - 2.39i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-10.2 - 1.80i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (6.74 + 8.03i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-8.93 - 1.57i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (7.25 - 8.64i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (0.0904 + 0.512i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-3.99 - 1.45i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (0.598 + 0.217i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-1.07 - 0.618i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-12.6 + 4.58i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-3.79 + 3.18i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.60090728877917315132690271351, −10.74757730036171864022995419432, −9.540977082841441382796028975899, −8.526671244579838074943113609710, −7.65635402910980297906364058592, −6.50082028757251007494073613357, −5.83815127411365432160185951017, −4.32196010604129023908792558188, −4.10070017139581588856003664297, −2.44777215224465053015660384428,
0.925253721746617780085437034521, 2.62050033430292919812307142905, 3.67060046464653720006716082323, 5.02067395045073111133306657538, 5.82390012728371549903669334673, 6.84184892080758195917006565118, 7.911286980649881714629663218406, 8.908314818605448511515363930839, 10.40557293275302155499818677193, 10.73238799299181552026128861904
