L(s) = 1 | + (−0.930 − 0.365i)2-s + (1.60 − 0.650i)3-s + (0.733 + 0.680i)4-s + (−0.128 − 0.0879i)5-s + (−1.73 + 0.0192i)6-s + (−2.44 − 1.01i)7-s + (−0.433 − 0.900i)8-s + (2.15 − 2.08i)9-s + (0.0879 + 0.128i)10-s + (−0.535 − 3.54i)11-s + (1.61 + 0.614i)12-s + (4.78 − 3.81i)13-s + (1.90 + 1.83i)14-s + (−0.264 − 0.0572i)15-s + (0.0747 + 0.997i)16-s + (1.95 + 0.602i)17-s + ⋯ |
L(s) = 1 | + (−0.658 − 0.258i)2-s + (0.926 − 0.375i)3-s + (0.366 + 0.340i)4-s + (−0.0576 − 0.0393i)5-s + (−0.707 + 0.00784i)6-s + (−0.923 − 0.382i)7-s + (−0.153 − 0.318i)8-s + (0.717 − 0.696i)9-s + (0.0278 + 0.0407i)10-s + (−0.161 − 1.07i)11-s + (0.467 + 0.177i)12-s + (1.32 − 1.05i)13-s + (0.509 + 0.490i)14-s + (−0.0682 − 0.0147i)15-s + (0.0186 + 0.249i)16-s + (0.473 + 0.146i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.946626 - 0.734003i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.946626 - 0.734003i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.930 + 0.365i)T \) |
| 3 | \( 1 + (-1.60 + 0.650i)T \) |
| 7 | \( 1 + (2.44 + 1.01i)T \) |
good | 5 | \( 1 + (0.128 + 0.0879i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (0.535 + 3.54i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (-4.78 + 3.81i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-1.95 - 0.602i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (2.10 - 1.21i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.62 - 8.49i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (-4.78 + 1.09i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-1.73 - 1.00i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.28 - 2.11i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (9.06 - 4.36i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (4.79 + 2.30i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (2.30 - 5.86i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (-0.683 + 0.736i)T + (-3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (-5.97 + 4.07i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (-0.373 - 0.403i)T + (-4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (6.76 - 11.7i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.13 - 1.17i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-1.20 + 0.473i)T + (53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (-8.15 - 14.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (9.06 - 11.3i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (0.649 + 0.0978i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 + 2.73iT - 97T^{2} \) |
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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
−11.49036278848950301840126226187, −10.37355892730515592853548989900, −9.734893003589385280117811528181, −8.466616314735438623589071263904, −8.165454483872630511285639004202, −6.86511506735979496602085604183, −5.91137122629734493765160275724, −3.64428322281689696235531682710, −3.06175695709396295148905501127, −1.10035078193047630491548897608,
2.03299610736499580412360759926, 3.41581763839715301421396832945, 4.75664092335261142210876659410, 6.42300219662143502105750984022, 7.15107476876417607514831463542, 8.506978107157854808152976900677, 8.986472248315944985069884331280, 9.937480817590448197609685828888, 10.62550192847618302728551724838, 11.94603851716426564907527535489