| L(s) = 1 | + (−0.728 − 1.21i)2-s + (1.02 + 1.39i)3-s + (−0.939 + 1.76i)4-s + (−0.779 + 2.14i)5-s + (0.948 − 2.25i)6-s + (0.984 + 0.173i)7-s + (2.82 − 0.147i)8-s + (−0.904 + 2.86i)9-s + (3.16 − 0.614i)10-s + (5.69 − 2.07i)11-s + (−3.42 + 0.494i)12-s + (2.72 − 2.28i)13-s + (−0.506 − 1.32i)14-s + (−3.78 + 1.10i)15-s + (−2.23 − 3.31i)16-s + (−5.84 + 3.37i)17-s + ⋯ |
| L(s) = 1 | + (−0.514 − 0.857i)2-s + (0.590 + 0.806i)3-s + (−0.469 + 0.882i)4-s + (−0.348 + 0.957i)5-s + (0.387 − 0.921i)6-s + (0.372 + 0.0656i)7-s + (0.998 − 0.0519i)8-s + (−0.301 + 0.953i)9-s + (1.00 − 0.194i)10-s + (1.71 − 0.624i)11-s + (−0.989 + 0.142i)12-s + (0.755 − 0.634i)13-s + (−0.135 − 0.352i)14-s + (−0.978 + 0.284i)15-s + (−0.558 − 0.829i)16-s + (−1.41 + 0.818i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.539 - 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.539 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.22648 + 0.670409i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22648 + 0.670409i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.728 + 1.21i)T \) |
| 3 | \( 1 + (-1.02 - 1.39i)T \) |
| 7 | \( 1 + (-0.984 - 0.173i)T \) |
| good | 5 | \( 1 + (0.779 - 2.14i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (-5.69 + 2.07i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.72 + 2.28i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (5.84 - 3.37i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.21 - 0.701i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.430 - 2.44i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (1.90 - 2.26i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.27 + 0.225i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-4.61 - 7.99i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.27 + 1.52i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (2.90 + 7.99i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.14 - 6.50i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 5.43iT - 53T^{2} \) |
| 59 | \( 1 + (12.9 + 4.70i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.52 + 8.64i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.52 - 4.19i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.92 - 3.34i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (1.77 - 3.07i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.893 + 1.06i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (6.03 + 5.06i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (11.5 + 6.66i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.40 + 2.69i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.60736082547343870015437715128, −9.661956491079449535503192111574, −8.797004015603404467131740631662, −8.362278350446405684862899032841, −7.26657262953534968689831092701, −6.15205453431030127843927366766, −4.54506366699718430649896624687, −3.66822815259772804910412707254, −3.10357630305339891408748122025, −1.64743269915817684218398692307,
0.886664155663924990420599456575, 1.92160162247515833936197380233, 4.05952144474924728593684137816, 4.70390365824821636212103465573, 6.22360964518650822899977790072, 6.80354913872862819719433740676, 7.62837589498075932669185819958, 8.664289005304547598757414458478, 8.969050916465076145329741668724, 9.616149246689725954700448846330
