| L(s) = 1 | + (−1.46 + 0.706i)2-s + (−3.52 − 4.41i)3-s + (−3.33 + 4.17i)4-s + (−12.5 + 6.03i)5-s + (8.29 + 3.99i)6-s + (−8.89 − 11.1i)7-s + (4.83 − 21.1i)8-s + (−1.10 + 4.83i)9-s + (14.1 − 17.7i)10-s + (12.1 + 53.3i)11-s + 30.2·12-s + (−3.30 − 14.4i)13-s + (20.9 + 10.0i)14-s + (70.8 + 34.1i)15-s + (−1.63 − 7.15i)16-s − 118.·17-s + ⋯ |
| L(s) = 1 | + (−0.518 + 0.249i)2-s + (−0.678 − 0.850i)3-s + (−0.416 + 0.522i)4-s + (−1.12 + 0.539i)5-s + (0.564 + 0.271i)6-s + (−0.480 − 0.602i)7-s + (0.213 − 0.936i)8-s + (−0.0408 + 0.179i)9-s + (0.446 − 0.560i)10-s + (0.333 + 1.46i)11-s + 0.727·12-s + (−0.0705 − 0.309i)13-s + (0.399 + 0.192i)14-s + (1.21 + 0.587i)15-s + (−0.0255 − 0.111i)16-s − 1.68·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0563i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0563i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.000255661 - 0.00907462i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.000255661 - 0.00907462i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (128. - 88.2i)T \) |
| good | 2 | \( 1 + (1.46 - 0.706i)T + (4.98 - 6.25i)T^{2} \) |
| 3 | \( 1 + (3.52 + 4.41i)T + (-6.00 + 26.3i)T^{2} \) |
| 5 | \( 1 + (12.5 - 6.03i)T + (77.9 - 97.7i)T^{2} \) |
| 7 | \( 1 + (8.89 + 11.1i)T + (-76.3 + 334. i)T^{2} \) |
| 11 | \( 1 + (-12.1 - 53.3i)T + (-1.19e3 + 577. i)T^{2} \) |
| 13 | \( 1 + (3.30 + 14.4i)T + (-1.97e3 + 953. i)T^{2} \) |
| 17 | \( 1 + 118.T + 4.91e3T^{2} \) |
| 19 | \( 1 + (-38.2 + 47.9i)T + (-1.52e3 - 6.68e3i)T^{2} \) |
| 23 | \( 1 + (-5.09 - 2.45i)T + (7.58e3 + 9.51e3i)T^{2} \) |
| 31 | \( 1 + (142. - 68.6i)T + (1.85e4 - 2.32e4i)T^{2} \) |
| 37 | \( 1 + (-92.5 + 405. i)T + (-4.56e4 - 2.19e4i)T^{2} \) |
| 41 | \( 1 + 4.49T + 6.89e4T^{2} \) |
| 43 | \( 1 + (-180. - 87.0i)T + (4.95e4 + 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-33.7 - 147. i)T + (-9.35e4 + 4.50e4i)T^{2} \) |
| 53 | \( 1 + (-80.8 + 38.9i)T + (9.28e4 - 1.16e5i)T^{2} \) |
| 59 | \( 1 - 73.5T + 2.05e5T^{2} \) |
| 61 | \( 1 + (31.8 + 39.8i)T + (-5.05e4 + 2.21e5i)T^{2} \) |
| 67 | \( 1 + (188. - 826. i)T + (-2.70e5 - 1.30e5i)T^{2} \) |
| 71 | \( 1 + (193. + 845. i)T + (-3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (542. + 261. i)T + (2.42e5 + 3.04e5i)T^{2} \) |
| 79 | \( 1 + (140. - 617. i)T + (-4.44e5 - 2.13e5i)T^{2} \) |
| 83 | \( 1 + (-531. + 666. i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (1.08e3 - 521. i)T + (4.39e5 - 5.51e5i)T^{2} \) |
| 97 | \( 1 + (1.06e3 - 1.34e3i)T + (-2.03e5 - 8.89e5i)T^{2} \) |
| 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
−16.25473055357298022611392322110, −15.08331839781004060936514050235, −13.15009839677231046872293751545, −12.35493988370822382807346493739, −11.06094431444034572175072136213, −9.283831502005562976337720708461, −7.32600950463501438645182316411, −7.03533203884847079476475349285, −4.05457026469857894388972143279, −0.01098724088555078686827738707,
4.27207754359396373457912904765, 5.77475078465079860041297911768, 8.405430781761440642134804652663, 9.430832151612638757130246694908, 10.96438601265951622909085759351, 11.66403694145427441601322714872, 13.50459655359470917759713944286, 15.23142320969710080308018710146, 16.12093640569523941350728738589, 16.98739416673421752686889717031
