L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s − 0.999i·8-s + (0.866 − 0.5i)9-s + 0.999i·10-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)16-s + (−0.133 + 0.5i)17-s − 0.999·18-s + (0.499 − 0.866i)20-s + (−0.499 + 0.866i)25-s + (−0.866 + 0.499i)26-s + (−0.866 − 0.5i)29-s + (0.866 − 0.499i)32-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s − 0.999i·8-s + (0.866 − 0.5i)9-s + 0.999i·10-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)16-s + (−0.133 + 0.5i)17-s − 0.999·18-s + (0.499 − 0.866i)20-s + (−0.499 + 0.866i)25-s + (−0.866 + 0.499i)26-s + (−0.866 − 0.5i)29-s + (0.866 − 0.499i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5070188846\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5070188846\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
good | 3 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (0.133 - 0.5i)T + (-0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 - 1.86i)T + (-0.866 + 0.5i)T^{2} \) |
| 43 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-1.36 - 1.36i)T + iT^{2} \) |
| 59 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 73 | \( 1 + iT - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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
−12.01407581032013662842355993193, −11.05988163799508652571986098476, −10.06184315309397580437611622674, −9.211754476680937660618560805775, −8.264503323948510568648088003993, −7.51861521370992103386934319526, −6.20234723941787485909099373062, −4.49174427699815411320582202086, −3.37499013462868163067109286651, −1.34807775428099862667202992747,
2.09578694339636394321536963084, 3.97760477020781213913853196707, 5.51791643397078146658073177094, 6.94721454909328462988678766020, 7.24701024201610192677332175876, 8.483813360022096736312784346088, 9.483757185573347529627337062633, 10.51877103205437787056288471915, 11.09736943488284385373993430078, 12.09473832027174785581575637604