| L(s) = 1 | + (−0.996 − 0.0871i)2-s + (0.984 + 0.173i)4-s + (0.702 − 2.12i)5-s + (−0.580 − 0.406i)7-s + (−0.965 − 0.258i)8-s + (−0.885 + 2.05i)10-s + (−0.183 + 0.502i)11-s + (−0.260 − 2.97i)13-s + (0.542 + 0.455i)14-s + (0.939 + 0.342i)16-s + (−1.53 + 0.410i)17-s + (4.21 − 2.43i)19-s + (1.06 − 1.96i)20-s + (0.226 − 0.485i)22-s + (−3.34 − 4.77i)23-s + ⋯ |
| L(s) = 1 | + (−0.704 − 0.0616i)2-s + (0.492 + 0.0868i)4-s + (0.314 − 0.949i)5-s + (−0.219 − 0.153i)7-s + (−0.341 − 0.0915i)8-s + (−0.279 + 0.649i)10-s + (−0.0551 + 0.151i)11-s + (−0.0722 − 0.825i)13-s + (0.145 + 0.121i)14-s + (0.234 + 0.0855i)16-s + (−0.371 + 0.0995i)17-s + (0.967 − 0.558i)19-s + (0.237 − 0.440i)20-s + (0.0482 − 0.103i)22-s + (−0.697 − 0.996i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.516 + 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.516 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.415971 - 0.737194i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.415971 - 0.737194i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.996 + 0.0871i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.702 + 2.12i)T \) |
| good | 7 | \( 1 + (0.580 + 0.406i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (0.183 - 0.502i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.260 + 2.97i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (1.53 - 0.410i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-4.21 + 2.43i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.34 + 4.77i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (3.54 - 2.97i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.226 + 1.28i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.90 - 7.10i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (3.75 - 4.47i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (3.29 + 7.06i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-6.41 + 9.15i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (4.01 - 4.01i)T - 53iT^{2} \) |
| 59 | \( 1 + (-6.74 + 2.45i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.669 + 3.79i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (15.3 - 1.34i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (2.33 + 1.35i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.25 + 12.1i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (2.08 + 2.48i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.15 + 13.1i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-7.06 - 12.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.89 - 2.74i)T + (62.3 - 74.3i)T^{2} \) |
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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
−9.960357145889865687682216218904, −9.086123991322764653757948753755, −8.410719410247119638235024463951, −7.58625573655268628052972258413, −6.57394911826871767845851633963, −5.55953516476717259609383695583, −4.64836472123196444824965570034, −3.27237172816809269831502251174, −1.92952518611424290775174398050, −0.51624592187265273022369716241,
1.71985164801931654131027723797, 2.85178743679064091527378166310, 3.96640178245329065392539520682, 5.58082481444069204343292628052, 6.27549014194490035402897678775, 7.24424483687230850523077696231, 7.82014076840826862393816646093, 9.096411609120817147687432363498, 9.630515958930258229413392483644, 10.38023816287793086672699585286
