L(s) = 1 | + (1 + 1.73i)2-s + (−0.5 + 0.866i)3-s + (−0.999 + 1.73i)4-s + (0.5 + 0.866i)5-s − 1.99·6-s + (1 + 1.73i)9-s + (−0.999 + 1.73i)10-s + (−0.5 + 0.866i)11-s + (−1 − 1.73i)12-s − 4·13-s − 0.999·15-s + (1.99 + 3.46i)16-s + (−1 + 1.73i)17-s + (−2 + 3.46i)18-s − 1.99·20-s + ⋯ |
L(s) = 1 | + (0.707 + 1.22i)2-s + (−0.288 + 0.499i)3-s + (−0.499 + 0.866i)4-s + (0.223 + 0.387i)5-s − 0.816·6-s + (0.333 + 0.577i)9-s + (−0.316 + 0.547i)10-s + (−0.150 + 0.261i)11-s + (−0.288 − 0.499i)12-s − 1.10·13-s − 0.258·15-s + (0.499 + 0.866i)16-s + (−0.242 + 0.420i)17-s + (−0.471 + 0.816i)18-s − 0.447·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.241450 + 1.89471i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.241450 + 1.89471i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-1 - 1.73i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.5 + 4.33i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6 - 10.3i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 + (-2 + 3.46i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (-7.5 - 12.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.06721126792996225281890271758, −10.32467625614385717755264612114, −9.571415605014629107687188324938, −8.162741939147974914382345281614, −7.42275819568418729090796066271, −6.60237174557722222786844192066, −5.63578811875978607975840195731, −4.82097633270886272408152654355, −4.07236695748109229217440871245, −2.32601443659742790056971909499,
0.968236289550503584798729107264, 2.27302924596790538133867218119, 3.41957232484765275166904434092, 4.63122778170841813451460445300, 5.39455014640119999984840278926, 6.71699091693613544161400637970, 7.56645059217730979516684969241, 8.931063904650405592360426580297, 9.824064317429458425643783876155, 10.58260609971970333494181149261