| L(s) = 1 | + (1.21 − 5.33i)2-s + (−5.18 + 6.49i)3-s + (−19.7 − 9.51i)4-s + (−3.70 + 4.64i)5-s + (28.3 + 35.5i)6-s + (−12.4 − 13.7i)7-s + (−47.5 + 59.6i)8-s + (−9.36 − 41.0i)9-s + (20.2 + 25.4i)10-s + (6.97 − 30.5i)11-s + (164. − 79.1i)12-s + (3.78 − 16.5i)13-s + (−88.3 + 49.6i)14-s + (−10.9 − 48.1i)15-s + (150. + 188. i)16-s + (−50.3 + 24.2i)17-s + ⋯ |
| L(s) = 1 | + (0.430 − 1.88i)2-s + (−0.997 + 1.25i)3-s + (−2.47 − 1.18i)4-s + (−0.331 + 0.415i)5-s + (1.92 + 2.41i)6-s + (−0.671 − 0.740i)7-s + (−2.10 + 2.63i)8-s + (−0.346 − 1.51i)9-s + (0.640 + 0.803i)10-s + (0.191 − 0.838i)11-s + (3.95 − 1.90i)12-s + (0.0807 − 0.353i)13-s + (−1.68 + 0.948i)14-s + (−0.189 − 0.828i)15-s + (2.35 + 2.95i)16-s + (−0.718 + 0.345i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.122215 + 0.233913i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.122215 + 0.233913i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (12.4 + 13.7i)T \) |
| good | 2 | \( 1 + (-1.21 + 5.33i)T + (-7.20 - 3.47i)T^{2} \) |
| 3 | \( 1 + (5.18 - 6.49i)T + (-6.00 - 26.3i)T^{2} \) |
| 5 | \( 1 + (3.70 - 4.64i)T + (-27.8 - 121. i)T^{2} \) |
| 11 | \( 1 + (-6.97 + 30.5i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (-3.78 + 16.5i)T + (-1.97e3 - 953. i)T^{2} \) |
| 17 | \( 1 + (50.3 - 24.2i)T + (3.06e3 - 3.84e3i)T^{2} \) |
| 19 | \( 1 + 16.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + (75.6 + 36.4i)T + (7.58e3 + 9.51e3i)T^{2} \) |
| 29 | \( 1 + (-90.4 + 43.5i)T + (1.52e4 - 1.90e4i)T^{2} \) |
| 31 | \( 1 + 153.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-88.5 + 42.6i)T + (3.15e4 - 3.96e4i)T^{2} \) |
| 41 | \( 1 + (-205. + 258. i)T + (-1.53e4 - 6.71e4i)T^{2} \) |
| 43 | \( 1 + (212. + 266. i)T + (-1.76e4 + 7.75e4i)T^{2} \) |
| 47 | \( 1 + (17.7 - 77.8i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (50.6 + 24.3i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 + (122. + 153. i)T + (-4.57e4 + 2.00e5i)T^{2} \) |
| 61 | \( 1 + (500. - 240. i)T + (1.41e5 - 1.77e5i)T^{2} \) |
| 67 | \( 1 + 379.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (945. + 455. i)T + (2.23e5 + 2.79e5i)T^{2} \) |
| 73 | \( 1 + (-267. - 1.17e3i)T + (-3.50e5 + 1.68e5i)T^{2} \) |
| 79 | \( 1 - 937.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (133. + 583. i)T + (-5.15e5 + 2.48e5i)T^{2} \) |
| 89 | \( 1 + (220. + 963. i)T + (-6.35e5 + 3.05e5i)T^{2} \) |
| 97 | \( 1 + 890.T + 9.12e5T^{2} \) |
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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
−14.05628139039882043349260567909, −12.86233013924496717201286068704, −11.60857221493280739888667435431, −10.79408414285489786008452344675, −10.28691640822391588303533487356, −9.078540562740522142708678498634, −5.87210721772776262241527159291, −4.32911590015820405253210463822, −3.41697463349790751200787566889, −0.19702796916170479646461275552,
4.70082273974654956884269764021, 6.06331346206750192396754325109, 6.78647540700361242488812254327, 7.931055804509712335035542109891, 9.265835225073395305551379660070, 11.98263291495733024469647208477, 12.66758676202956482611544797339, 13.51712007323988607004315322058, 14.91613484966879725342911982471, 16.06484959313204357369367922326
