| L(s) = 1 | + (0.761 + 2.34i)2-s + (0.433 + 0.314i)3-s + (1.55 − 1.13i)4-s + (−0.474 + 1.46i)5-s + (−0.408 + 1.25i)6-s + (22.8 − 16.6i)7-s + (19.7 + 14.3i)8-s + (−8.25 − 25.4i)9-s − 3.78·10-s + 1.03·12-s + (21.1 + 65.1i)13-s + (56.3 + 40.9i)14-s + (−0.665 + 0.483i)15-s + (−13.8 + 42.6i)16-s + (17.1 − 52.6i)17-s + (53.2 − 38.6i)18-s + ⋯ |
| L(s) = 1 | + (0.269 + 0.828i)2-s + (0.0834 + 0.0606i)3-s + (0.194 − 0.141i)4-s + (−0.0424 + 0.130i)5-s + (−0.0277 + 0.0854i)6-s + (1.23 − 0.896i)7-s + (0.874 + 0.635i)8-s + (−0.305 − 0.940i)9-s − 0.119·10-s + 0.0248·12-s + (0.451 + 1.38i)13-s + (1.07 + 0.781i)14-s + (−0.0114 + 0.00832i)15-s + (−0.216 + 0.666i)16-s + (0.244 − 0.751i)17-s + (0.697 − 0.506i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.780 - 0.625i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.24476 + 0.788840i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.24476 + 0.788840i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 + (-0.761 - 2.34i)T + (-6.47 + 4.70i)T^{2} \) |
| 3 | \( 1 + (-0.433 - 0.314i)T + (8.34 + 25.6i)T^{2} \) |
| 5 | \( 1 + (0.474 - 1.46i)T + (-101. - 73.4i)T^{2} \) |
| 7 | \( 1 + (-22.8 + 16.6i)T + (105. - 326. i)T^{2} \) |
| 13 | \( 1 + (-21.1 - 65.1i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-17.1 + 52.6i)T + (-3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-44.6 - 32.4i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + 178.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (91.5 - 66.5i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-21.8 - 67.3i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-170. + 123. i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (155. + 112. i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 + 208.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (414. + 301. i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (116. + 357. i)T + (-1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-409. + 297. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (144. - 445. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + 289.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (121. - 374. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (234. - 170. i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (52.4 + 161. i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (93.7 - 288. i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + 1.14e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-198. - 610. i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
| 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
−13.62594245033647129574322801211, −11.76954064136623985209184739642, −11.24468411592897830081265587859, −9.953914776838384071471536612978, −8.531697415743033864581459583222, −7.41344733563096570115877718512, −6.55822549626630575578777372652, −5.20473022830609273629745318385, −3.94195863896206658879920245860, −1.55640939933658983026900252100,
1.68552005827987984065579145854, 2.91574566675238578386163322742, 4.62743183239287270762939373154, 5.83832683941276195369074405023, 7.87774800040092181253916027800, 8.262698244320549803456348057781, 10.10157391374772308245325339834, 11.04431934332048960070337200749, 11.74903263285250179064740472679, 12.72122153734520238397374180764
