| L(s) = 1 | + (3.30 + 2.40i)2-s + (0.900 − 5.11i)3-s + (2.69 + 8.29i)4-s + (3.98 + 5.48i)5-s + (15.2 − 14.7i)6-s + (−17.8 + 5.81i)7-s + (−0.905 + 2.78i)8-s + (−25.3 − 9.21i)9-s + 27.7i·10-s + (−27.8 + 23.5i)11-s + (44.8 − 6.32i)12-s + (24.8 − 34.2i)13-s + (−73.1 − 23.7i)14-s + (31.6 − 15.4i)15-s + (46.7 − 33.9i)16-s + (16.8 − 12.2i)17-s + ⋯ |
| L(s) = 1 | + (1.16 + 0.849i)2-s + (0.173 − 0.984i)3-s + (0.336 + 1.03i)4-s + (0.356 + 0.490i)5-s + (1.03 − 1.00i)6-s + (−0.965 + 0.313i)7-s + (−0.0400 + 0.123i)8-s + (−0.939 − 0.341i)9-s + 0.877i·10-s + (−0.763 + 0.646i)11-s + (1.07 − 0.152i)12-s + (0.530 − 0.730i)13-s + (−1.39 − 0.453i)14-s + (0.545 − 0.266i)15-s + (0.730 − 0.530i)16-s + (0.240 − 0.174i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.02789 + 0.473373i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.02789 + 0.473373i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.900 + 5.11i)T \) |
| 11 | \( 1 + (27.8 - 23.5i)T \) |
| good | 2 | \( 1 + (-3.30 - 2.40i)T + (2.47 + 7.60i)T^{2} \) |
| 5 | \( 1 + (-3.98 - 5.48i)T + (-38.6 + 118. i)T^{2} \) |
| 7 | \( 1 + (17.8 - 5.81i)T + (277. - 201. i)T^{2} \) |
| 13 | \( 1 + (-24.8 + 34.2i)T + (-678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-16.8 + 12.2i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-4.70 - 1.52i)T + (5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 - 209. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (69.0 + 212. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-195. - 142. i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-51.4 - 158. i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (25.5 - 78.6i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + 255. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (136. + 44.2i)T + (8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-269. + 370. i)T + (-4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (264. - 85.9i)T + (1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (101. + 139. i)T + (-7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 - 110.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (518. + 714. i)T + (-1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (664. - 215. i)T + (3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-266. + 366. i)T + (-1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (1.04e3 - 757. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 - 924. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (-252. - 183. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.85071627369371468483834355447, −15.05441130670636101199234711268, −13.66820823367686260649471752388, −13.17479412860449414789933083632, −12.04152275730450625770734560010, −9.935459740427492344681974411716, −7.81345382338378987410303587619, −6.60626711592425967015838006208, −5.60487380540124395018965594637, −3.08006093671413596884190535327,
3.02204265459841141505126175583, 4.43677347052914211530753254511, 5.87442068381224005100424496692, 8.721487023326445142343369605269, 10.18111243688317335130390153284, 11.16657025086179579991933588199, 12.70650491021046526692266607683, 13.57065959712633941006672157474, 14.61085104562866799271940613497, 16.09997238386053753487829873861
