| L(s) = 1 | + (−0.347 − 1.96i)2-s + (6.25 − 5.24i)3-s + (−3.75 + 1.36i)4-s + (0.852 + 0.310i)5-s + (−12.5 − 10.4i)6-s + (−4.68 + 8.10i)7-s + (4 + 6.92i)8-s + (6.87 − 38.9i)9-s + (0.315 − 1.78i)10-s + (−15.6 − 27.1i)11-s + (−16.3 + 28.2i)12-s + (54.0 + 45.3i)13-s + (17.5 + 6.40i)14-s + (6.95 − 2.53i)15-s + (12.2 − 10.2i)16-s + (17.1 + 97.1i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (1.20 − 1.00i)3-s + (−0.469 + 0.171i)4-s + (0.0762 + 0.0277i)5-s + (−0.850 − 0.713i)6-s + (−0.252 + 0.437i)7-s + (0.176 + 0.306i)8-s + (0.254 − 1.44i)9-s + (0.00996 − 0.0565i)10-s + (−0.428 − 0.742i)11-s + (−0.392 + 0.680i)12-s + (1.15 + 0.966i)13-s + (0.335 + 0.122i)14-s + (0.119 − 0.0435i)15-s + (0.191 − 0.160i)16-s + (0.244 + 1.38i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0478 + 0.998i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0478 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.14191 - 1.08850i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14191 - 1.08850i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.347 + 1.96i)T \) |
| 19 | \( 1 + (70.3 - 43.6i)T \) |
| good | 3 | \( 1 + (-6.25 + 5.24i)T + (4.68 - 26.5i)T^{2} \) |
| 5 | \( 1 + (-0.852 - 0.310i)T + (95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (4.68 - 8.10i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (15.6 + 27.1i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-54.0 - 45.3i)T + (381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-17.1 - 97.1i)T + (-4.61e3 + 1.68e3i)T^{2} \) |
| 23 | \( 1 + (-150. + 54.9i)T + (9.32e3 - 7.82e3i)T^{2} \) |
| 29 | \( 1 + (-18.8 + 106. i)T + (-2.29e4 - 8.34e3i)T^{2} \) |
| 31 | \( 1 + (27.0 - 46.8i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 355.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (166. - 139. i)T + (1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (284. + 103. i)T + (6.09e4 + 5.11e4i)T^{2} \) |
| 47 | \( 1 + (-61.8 + 350. i)T + (-9.75e4 - 3.55e4i)T^{2} \) |
| 53 | \( 1 + (-46.9 + 17.1i)T + (1.14e5 - 9.56e4i)T^{2} \) |
| 59 | \( 1 + (29.1 + 165. i)T + (-1.92e5 + 7.02e4i)T^{2} \) |
| 61 | \( 1 + (751. - 273. i)T + (1.73e5 - 1.45e5i)T^{2} \) |
| 67 | \( 1 + (-69.7 + 395. i)T + (-2.82e5 - 1.02e5i)T^{2} \) |
| 71 | \( 1 + (-370. - 134. i)T + (2.74e5 + 2.30e5i)T^{2} \) |
| 73 | \( 1 + (-487. + 409. i)T + (6.75e4 - 3.83e5i)T^{2} \) |
| 79 | \( 1 + (-536. + 450. i)T + (8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (128. - 222. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (660. + 554. i)T + (1.22e5 + 6.94e5i)T^{2} \) |
| 97 | \( 1 + (-153. - 871. i)T + (-8.57e5 + 3.12e5i)T^{2} \) |
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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
−15.21748010859894967579839388559, −13.94481172948611403238229341129, −13.21718474795361949934092468041, −12.18263696639184824886035499420, −10.61905951309050243606875797197, −8.855315202891473834524466106132, −8.267319464448225977223591824945, −6.38377680080224155535426933115, −3.49753636714206762477403229488, −1.84987687167744423070858128494,
3.36763367383769799985431694910, 5.02782829709569474559626916607, 7.25148233835152174570278578733, 8.597847563296690131359352990981, 9.605125545169534187579810598308, 10.70813508913450925024876182106, 13.11890734665006425288572639370, 13.95555389595620595296872142260, 15.32379122253748436546387419664, 15.60796549944506082056247583904
