L(s) = 1 | + (−0.666 − 0.484i)2-s + (−0.296 + 0.913i)3-s + (−0.408 − 1.25i)4-s + (−2.41 + 1.75i)5-s + (0.640 − 0.465i)6-s + (−0.309 − 0.951i)7-s + (−0.845 + 2.60i)8-s + (1.68 + 1.22i)9-s + 2.45·10-s + 1.26·12-s + (1.78 + 1.29i)13-s + (−0.254 + 0.783i)14-s + (−0.886 − 2.72i)15-s + (−0.315 + 0.229i)16-s + (3.56 − 2.59i)17-s + (−0.528 − 1.62i)18-s + ⋯ |
L(s) = 1 | + (−0.471 − 0.342i)2-s + (−0.171 + 0.527i)3-s + (−0.204 − 0.628i)4-s + (−1.08 + 0.784i)5-s + (0.261 − 0.189i)6-s + (−0.116 − 0.359i)7-s + (−0.298 + 0.919i)8-s + (0.560 + 0.406i)9-s + 0.777·10-s + 0.366·12-s + (0.493 + 0.358i)13-s + (−0.0680 + 0.209i)14-s + (−0.228 − 0.704i)15-s + (−0.0789 + 0.0573i)16-s + (0.864 − 0.628i)17-s + (−0.124 − 0.383i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 + 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.898 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0362327 - 0.157094i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0362327 - 0.157094i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.309 + 0.951i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.666 + 0.484i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (0.296 - 0.913i)T + (-2.42 - 1.76i)T^{2} \) |
| 5 | \( 1 + (2.41 - 1.75i)T + (1.54 - 4.75i)T^{2} \) |
| 13 | \( 1 + (-1.78 - 1.29i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-3.56 + 2.59i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.533 + 1.64i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 8.39T + 23T^{2} \) |
| 29 | \( 1 + (1.01 + 3.13i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (6.04 + 4.39i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (2.71 + 8.35i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (1.66 - 5.12i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 9.44T + 43T^{2} \) |
| 47 | \( 1 + (1.66 - 5.13i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (7.60 + 5.52i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.07 - 3.30i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-10.5 + 7.63i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 4.32T + 67T^{2} \) |
| 71 | \( 1 + (3.56 - 2.58i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (4.55 + 14.0i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (5.81 + 4.22i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-6.17 + 4.48i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + (2.30 + 1.67i)T + (29.9 + 92.2i)T^{2} \) |
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
−9.954334543214873566970561981757, −9.317745655368642744054056907483, −8.052063632825600617443912346000, −7.48425626507179439994729736637, −6.38282289493389631915521784397, −5.29861866769133794330761126499, −4.25226883681189601109513191975, −3.46073417090502561351592897910, −1.89107253354865290435878085356, −0.10137829690085258235554185073,
1.39402643690917262808093111217, 3.50276530104497930148179988745, 3.98772864778399415561186909274, 5.35123513843522143036813390296, 6.48914937232379297820454535770, 7.32847036945320314613250408382, 8.247830113775132372723445209259, 8.378926948085799403814689447619, 9.541589748368930793030104606812, 10.35341010178266757597247538874