L(s) = 1 | + 0.714i·2-s + (0.852 − 1.50i)3-s + 1.48·4-s + 0.589·5-s + (1.07 + 0.609i)6-s + (2.38 − 1.14i)7-s + 2.49i·8-s + (−1.54 − 2.57i)9-s + 0.420i·10-s + 3.80i·11-s + (1.27 − 2.24i)12-s + 2.33i·13-s + (0.820 + 1.70i)14-s + (0.502 − 0.888i)15-s + 1.19·16-s − 1.72·17-s + ⋯ |
L(s) = 1 | + 0.505i·2-s + (0.492 − 0.870i)3-s + 0.744·4-s + 0.263·5-s + (0.439 + 0.248i)6-s + (0.900 − 0.433i)7-s + 0.881i·8-s + (−0.514 − 0.857i)9-s + 0.133i·10-s + 1.14i·11-s + (0.366 − 0.648i)12-s + 0.646i·13-s + (0.219 + 0.455i)14-s + (0.129 − 0.229i)15-s + 0.299·16-s − 0.419·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0660i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.17597 - 0.0719389i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.17597 - 0.0719389i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.852 + 1.50i)T \) |
| 7 | \( 1 + (-2.38 + 1.14i)T \) |
| 23 | \( 1 + iT \) |
good | 2 | \( 1 - 0.714iT - 2T^{2} \) |
| 5 | \( 1 - 0.589T + 5T^{2} \) |
| 11 | \( 1 - 3.80iT - 11T^{2} \) |
| 13 | \( 1 - 2.33iT - 13T^{2} \) |
| 17 | \( 1 + 1.72T + 17T^{2} \) |
| 19 | \( 1 + 5.00iT - 19T^{2} \) |
| 29 | \( 1 + 2.26iT - 29T^{2} \) |
| 31 | \( 1 + 0.103iT - 31T^{2} \) |
| 37 | \( 1 + 3.37T + 37T^{2} \) |
| 41 | \( 1 + 2.26T + 41T^{2} \) |
| 43 | \( 1 + 3.75T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + 6.67iT - 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 8.82iT - 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 - 7.24iT - 71T^{2} \) |
| 73 | \( 1 - 8.17iT - 73T^{2} \) |
| 79 | \( 1 + 1.35T + 79T^{2} \) |
| 83 | \( 1 + 17.1T + 83T^{2} \) |
| 89 | \( 1 + 2.13T + 89T^{2} \) |
| 97 | \( 1 + 8.67iT - 97T^{2} \) |
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
−11.23169166306779223564391374332, −10.04086309528908526545979709075, −8.905688543590731216574502777308, −8.035365570579815297554612892334, −7.13890073807824520978213871750, −6.78987360104334287115300636013, −5.52821828219243523205329112324, −4.27773637087665980945970097262, −2.48095794455451942423453657294, −1.68628653719373395172003888227,
1.78451376324322010562160764934, 2.95099933350794807399325696094, 3.89797075719592689481857328784, 5.33312824177208999474623364576, 6.07529703476786688046037662181, 7.66524760175401423782251760324, 8.361166006348400773449781872553, 9.319219175630751237887926667638, 10.35493542967462467689261359925, 10.86959783899189778938116736080