| L(s) = 1 | + (−0.431 + 2.03i)2-s + (1.72 + 1.91i)3-s + (−2.11 − 0.942i)4-s + 3.56i·5-s + (−4.63 + 2.67i)6-s + (1.47 − 3.31i)7-s + (0.386 − 0.531i)8-s + (−0.378 + 3.60i)9-s + (−7.23 − 1.53i)10-s + (−1.56 + 0.164i)11-s + (−1.84 − 5.67i)12-s + (2.76 − 2.31i)13-s + (6.10 + 4.43i)14-s + (−6.81 + 6.13i)15-s + (−2.18 − 2.42i)16-s + (−0.247 + 2.35i)17-s + ⋯ |
| L(s) = 1 | + (−0.305 + 1.43i)2-s + (0.994 + 1.10i)3-s + (−1.05 − 0.471i)4-s + 1.59i·5-s + (−1.89 + 1.09i)6-s + (0.557 − 1.25i)7-s + (0.136 − 0.188i)8-s + (−0.126 + 1.20i)9-s + (−2.28 − 0.486i)10-s + (−0.470 + 0.0494i)11-s + (−0.531 − 1.63i)12-s + (0.766 − 0.642i)13-s + (1.63 + 1.18i)14-s + (−1.75 + 1.58i)15-s + (−0.546 − 0.606i)16-s + (−0.0599 + 0.570i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.276i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.225509 - 1.59676i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.225509 - 1.59676i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-2.76 + 2.31i)T \) |
| 31 | \( 1 + (-1.85 - 5.24i)T \) |
| good | 2 | \( 1 + (0.431 - 2.03i)T + (-1.82 - 0.813i)T^{2} \) |
| 3 | \( 1 + (-1.72 - 1.91i)T + (-0.313 + 2.98i)T^{2} \) |
| 5 | \( 1 - 3.56iT - 5T^{2} \) |
| 7 | \( 1 + (-1.47 + 3.31i)T + (-4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (1.56 - 0.164i)T + (10.7 - 2.28i)T^{2} \) |
| 17 | \( 1 + (0.247 - 2.35i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (0.0995 + 0.0896i)T + (1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-8.30 + 3.69i)T + (15.3 - 17.0i)T^{2} \) |
| 29 | \( 1 + (7.06 + 1.50i)T + (26.4 + 11.7i)T^{2} \) |
| 37 | \( 1 + (-1.38 - 0.799i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.652 - 3.07i)T + (-37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-6.22 + 6.90i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (5.64 + 1.83i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.292 + 0.212i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.131 - 0.617i)T + (-53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (7.37 + 12.7i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.71 + 2.14i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (7.91 + 0.831i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (-5.10 - 7.02i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-9.19 - 6.68i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (10.3 - 3.37i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (9.66 - 1.01i)T + (87.0 - 18.5i)T^{2} \) |
| 97 | \( 1 + (5.32 - 11.9i)T + (-64.9 - 72.0i)T^{2} \) |
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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
−10.97958119846224000102094372807, −10.75993079184670840452775607725, −9.862822576337665230793168395477, −8.752982481204248729066139330568, −7.944288548363753982150402712080, −7.25038371358582066660819643052, −6.39534723624515695999611863576, −5.02641256042774242873895128223, −3.82959060738407267761707612553, −2.86874026435584371646622296785,
1.19500153204084735401558403738, 1.98451759152611030985297986309, 3.05262445065649979492167974587, 4.58391387197930834671042130996, 5.79506715798629754629077617273, 7.47525135090106104028927385165, 8.493029541338807149409473273804, 9.017469548989007276071293103331, 9.391940863143798180300456190378, 11.16460060137392807316370124077
