L(s) = 1 | + (−0.309 + 0.535i)2-s + (−0.309 + 0.535i)3-s + (0.309 + 0.535i)4-s + (−0.190 − 0.330i)6-s + (0.809 + 1.40i)7-s − 0.999·8-s + (0.309 + 0.535i)9-s + (−0.5 + 0.866i)11-s − 0.381·12-s − 14-s − 0.381·18-s + (−0.309 − 0.535i)19-s − 21-s + (−0.309 − 0.535i)22-s + (0.809 − 1.40i)23-s + (0.309 − 0.535i)24-s + ⋯ |
L(s) = 1 | + (−0.309 + 0.535i)2-s + (−0.309 + 0.535i)3-s + (0.309 + 0.535i)4-s + (−0.190 − 0.330i)6-s + (0.809 + 1.40i)7-s − 0.999·8-s + (0.309 + 0.535i)9-s + (−0.5 + 0.866i)11-s − 0.381·12-s − 14-s − 0.381·18-s + (−0.309 − 0.535i)19-s − 21-s + (−0.309 − 0.535i)22-s + (0.809 − 1.40i)23-s + (0.309 − 0.535i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9857074539\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9857074539\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \) |
| 3 | \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 0.618T + T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 - 0.618T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.61T + T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 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
−9.580483018680013414023873688358, −8.784114554784802576817259306832, −8.352639247408122757962299488481, −7.43166014973640479360378907400, −6.78186681193309813690363032348, −5.69600741792171630325817774177, −5.01957573549938067030827939472, −4.30452985394429020985037419782, −2.74961767110841020537892290697, −2.15467733325613833669887998447,
0.905109254228565189912969671719, 1.51528193674904816317104788736, 2.97560296740726181559893029783, 3.95294927102255024444808590702, 5.09530791055393373279590093659, 5.93855595167426022160699155345, 6.78348974292252778274107066961, 7.42491313442271485612047687194, 8.238265609573273482365474410219, 9.237639672395218809256543240484