L(s) = 1 | + (−0.774 + 1.18i)2-s + (−0.637 + 1.61i)3-s + (−0.801 − 1.83i)4-s + (0.686 + 0.396i)5-s + (−1.41 − 2.00i)6-s + (2.35 − 1.35i)7-s + (2.78 + 0.469i)8-s + (−2.18 − 2.05i)9-s + (−1 + 0.505i)10-s + (−1.71 − 2.96i)11-s + (3.46 − 0.121i)12-s + (−1.68 + 2.92i)13-s + (−0.213 + 3.83i)14-s + (−1.07 + 0.852i)15-s + (−2.71 + 2.93i)16-s − 2.52i·17-s + ⋯ |
L(s) = 1 | + (−0.547 + 0.836i)2-s + (−0.368 + 0.929i)3-s + (−0.400 − 0.916i)4-s + (0.306 + 0.177i)5-s + (−0.576 − 0.817i)6-s + (0.888 − 0.513i)7-s + (0.986 + 0.166i)8-s + (−0.728 − 0.684i)9-s + (−0.316 + 0.159i)10-s + (−0.516 − 0.894i)11-s + (0.999 − 0.0351i)12-s + (−0.467 + 0.809i)13-s + (−0.0570 + 1.02i)14-s + (−0.277 + 0.220i)15-s + (−0.678 + 0.734i)16-s − 0.612i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.166 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.166 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.427753 + 0.361501i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.427753 + 0.361501i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.774 - 1.18i)T \) |
| 3 | \( 1 + (0.637 - 1.61i)T \) |
good | 5 | \( 1 + (-0.686 - 0.396i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-2.35 + 1.35i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.71 + 2.96i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.68 - 2.92i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 2.52iT - 17T^{2} \) |
| 19 | \( 1 - 2.20iT - 19T^{2} \) |
| 23 | \( 1 + (1.07 - 1.86i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.686 - 0.396i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.47 + 0.852i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.74T + 37T^{2} \) |
| 41 | \( 1 + (-0.127 - 0.0737i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (6.01 - 3.47i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.77 - 10.0i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 8.51iT - 53T^{2} \) |
| 59 | \( 1 + (-2.58 + 4.48i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.68 + 2.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.01 + 3.47i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 1.75T + 71T^{2} \) |
| 73 | \( 1 + 2.37T + 73T^{2} \) |
| 79 | \( 1 + (-8.80 + 5.08i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.62 + 6.28i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 5.34iT - 89T^{2} \) |
| 97 | \( 1 + (-6.24 - 10.8i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−16.70985012382218553682260935486, −15.88673281793730288996005956496, −14.56503076456082401959329297435, −13.87243004290686964766825119737, −11.44385518035936395695076494494, −10.42897571371634572352597442852, −9.267143690196014933793692168026, −7.81937093685503095898891737418, −6.02210048266719320367213666561, −4.62632618248830518501790350783,
2.12226860007967066366062149714, 5.18938050780882079827077089902, 7.43637056954322732575378759682, 8.534285118964387069443086391372, 10.22233359038497806288468788253, 11.49255145271192746749948920204, 12.50552586011408585776297769771, 13.36656999084686644948595551036, 14.99755678490269295429074183774, 16.89506192121168170823699174585