L(s) = 1 | − 3-s + 4-s + (0.5 − 0.866i)5-s + (0.866 + 0.5i)7-s + 9-s − i·11-s − 12-s − i·13-s + (−0.5 + 0.866i)15-s + 16-s + (−0.866 + 0.5i)17-s + (−0.866 + 0.5i)19-s + (0.5 − 0.866i)20-s + (−0.866 − 0.5i)21-s + (−0.5 − 0.866i)23-s + ⋯ |
L(s) = 1 | − 3-s + 4-s + (0.5 − 0.866i)5-s + (0.866 + 0.5i)7-s + 9-s − i·11-s − 12-s − i·13-s + (−0.5 + 0.866i)15-s + 16-s + (−0.866 + 0.5i)17-s + (−0.866 + 0.5i)19-s + (0.5 − 0.866i)20-s + (−0.866 − 0.5i)21-s + (−0.5 − 0.866i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.156659713\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.156659713\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 11 | \( 1 + iT \) |
| 13 | \( 1 + iT \) |
good | 2 | \( 1 - T^{2} \) |
| 5 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - 2iT - T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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
−10.08234156202567264545958870343, −8.678923571276888362897515283937, −8.374952534294332309688940706512, −7.14897190275539500806440798400, −6.30103558602055852034410262809, −5.51910178544668644586195353598, −5.13088880962944760272783609645, −3.77672999528996976354180396143, −2.22919647374958718971932413918, −1.26727669923882418742744806176,
1.73903523031946344389680507450, 2.39450824484079272980002579970, 4.15857051072008364465171732540, 4.79929852761095887957714062624, 6.18879206060129602849664160179, 6.48248103895293987437629647669, 7.30368034853705107328550495421, 7.912041136976989248027801236434, 9.597394808135553687389880974757, 10.00338128166504996809591135666