L(s) = 1 | + (−1.96 + 0.349i)2-s + (−0.313 − 0.129i)3-s + (3.75 − 1.37i)4-s + (−3.56 − 8.61i)5-s + (0.663 + 0.146i)6-s + (−6.10 + 3.42i)7-s + (−6.91 + 4.02i)8-s + (−6.28 − 6.28i)9-s + (10.0 + 15.7i)10-s + (4.82 + 11.6i)11-s + (−1.35 − 0.0559i)12-s + (−2.75 + 6.66i)13-s + (10.8 − 8.88i)14-s + 3.16i·15-s + (12.2 − 10.3i)16-s + 28.4·17-s + ⋯ |
L(s) = 1 | + (−0.984 + 0.174i)2-s + (−0.104 − 0.0433i)3-s + (0.938 − 0.344i)4-s + (−0.713 − 1.72i)5-s + (0.110 + 0.0243i)6-s + (−0.871 + 0.489i)7-s + (−0.864 + 0.503i)8-s + (−0.698 − 0.698i)9-s + (1.00 + 1.57i)10-s + (0.438 + 1.05i)11-s + (−0.113 − 0.00465i)12-s + (−0.212 + 0.512i)13-s + (0.772 − 0.634i)14-s + 0.211i·15-s + (0.762 − 0.646i)16-s + 1.67·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.124978 + 0.190302i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.124978 + 0.190302i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.96 - 0.349i)T \) |
| 7 | \( 1 + (6.10 - 3.42i)T \) |
good | 3 | \( 1 + (0.313 + 0.129i)T + (6.36 + 6.36i)T^{2} \) |
| 5 | \( 1 + (3.56 + 8.61i)T + (-17.6 + 17.6i)T^{2} \) |
| 11 | \( 1 + (-4.82 - 11.6i)T + (-85.5 + 85.5i)T^{2} \) |
| 13 | \( 1 + (2.75 - 6.66i)T + (-119. - 119. i)T^{2} \) |
| 17 | \( 1 - 28.4T + 289T^{2} \) |
| 19 | \( 1 + (9.23 - 22.2i)T + (-255. - 255. i)T^{2} \) |
| 23 | \( 1 + (-7.97 - 7.97i)T + 529iT^{2} \) |
| 29 | \( 1 + (1.69 - 4.09i)T + (-594. - 594. i)T^{2} \) |
| 31 | \( 1 + 20.5iT - 961T^{2} \) |
| 37 | \( 1 + (48.6 - 20.1i)T + (968. - 968. i)T^{2} \) |
| 41 | \( 1 + (3.17 - 3.17i)T - 1.68e3iT^{2} \) |
| 43 | \( 1 + (-9.26 - 22.3i)T + (-1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + 62.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + (-10.5 - 25.3i)T + (-1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (31.4 + 76.0i)T + (-2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (55.6 + 23.0i)T + (2.63e3 + 2.63e3i)T^{2} \) |
| 67 | \( 1 + (24.5 - 59.3i)T + (-3.17e3 - 3.17e3i)T^{2} \) |
| 71 | \( 1 + (-33.2 + 33.2i)T - 5.04e3iT^{2} \) |
| 73 | \( 1 + (30.5 - 30.5i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 5.67iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (27.1 - 65.5i)T + (-4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (48.8 + 48.8i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 - 36.2iT - 9.40e3T^{2} \) |
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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
−12.18143965518850131504780144652, −11.69449323634785051257451262442, −9.892861681456845881602144252007, −9.353143859218448286728310628631, −8.528624544478125970200316841859, −7.61398727590445889752737449802, −6.28110903041286254830435231078, −5.22912969098922856409726270351, −3.56703609848187204903201745703, −1.41654402179624618586293763416,
0.17802067516859139114569897033, 2.89596015237367980868022387688, 3.40180000924373756622653650057, 5.94368657197495421390346986468, 6.90889197159305724822142264730, 7.65165615102133730055512826279, 8.704683953844786253125565739417, 10.12440093603574663292461331487, 10.68265472272451099625586576272, 11.31627988476144001607447076782