| L(s) = 1 | + (−2.44 + 1.41i)2-s + (−8.62 − 2.58i)3-s + (3.99 − 6.92i)4-s + (−2.28 + 1.32i)5-s + (24.7 − 5.85i)6-s + (47.8 + 10.3i)7-s + 22.6i·8-s + (67.6 + 44.5i)9-s + (3.73 − 6.47i)10-s + (99.2 + 57.2i)11-s + (−52.4 + 49.3i)12-s + 102.·13-s + (−131. + 42.3i)14-s + (23.1 − 5.47i)15-s + (−32.0 − 55.4i)16-s + (201. + 116. i)17-s + ⋯ |
| L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.957 − 0.287i)3-s + (0.249 − 0.433i)4-s + (−0.0915 + 0.0528i)5-s + (0.688 − 0.162i)6-s + (0.977 + 0.211i)7-s + 0.353i·8-s + (0.834 + 0.550i)9-s + (0.0373 − 0.0647i)10-s + (0.820 + 0.473i)11-s + (−0.363 + 0.342i)12-s + 0.606·13-s + (−0.673 + 0.216i)14-s + (0.102 − 0.0243i)15-s + (−0.125 − 0.216i)16-s + (0.698 + 0.403i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.461i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.887 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.911243 + 0.222772i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.911243 + 0.222772i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.44 - 1.41i)T \) |
| 3 | \( 1 + (8.62 + 2.58i)T \) |
| 7 | \( 1 + (-47.8 - 10.3i)T \) |
| good | 5 | \( 1 + (2.28 - 1.32i)T + (312.5 - 541. i)T^{2} \) |
| 11 | \( 1 + (-99.2 - 57.2i)T + (7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 - 102.T + 2.85e4T^{2} \) |
| 17 | \( 1 + (-201. - 116. i)T + (4.17e4 + 7.23e4i)T^{2} \) |
| 19 | \( 1 + (190. + 329. i)T + (-6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (-676. + 390. i)T + (1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 - 807. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + (-27.2 + 47.2i)T + (-4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + (-367. - 636. i)T + (-9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 - 1.81e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 3.30e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + (-523. + 302. i)T + (2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + (2.72e3 + 1.57e3i)T + (3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (2.90e3 + 1.67e3i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (3.38e3 + 5.86e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (3.08e3 - 5.35e3i)T + (-1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + 1.58e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + (-1.42e3 + 2.46e3i)T + (-1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-196. - 339. i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + 1.24e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 + (6.61e3 - 3.82e3i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 97 | \( 1 - 1.32e4T + 8.85e7T^{2} \) |
| 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
−15.50243345513404017209288018111, −14.47465774824854395859021201867, −12.80060867413969594585195550622, −11.49466184474921233052080926063, −10.72504093757998777094146710889, −9.067213595626322825365082869916, −7.60038053559783838683544305752, −6.34185664508420239861637107808, −4.82727376327431797718161935552, −1.34072768783800413180717870607,
1.11015897770328187290559082808, 4.09534986708229252672994639318, 5.92027198943299969205175851806, 7.61238284588144364358608846454, 9.122499151737902089053418347847, 10.55231250334989912358556361647, 11.40279541431311993614159624536, 12.31529544025109503185238473149, 14.02624385197705665744604128993, 15.47308334365514152801597673563
