L(s) = 1 | − i·2-s − 3-s − 4-s − i·5-s + i·6-s − 1.35i·7-s + i·8-s + 9-s − 10-s − 5.29i·11-s + 12-s − 1.35·14-s + i·15-s + 16-s + 2.04·17-s − i·18-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577·3-s − 0.5·4-s − 0.447i·5-s + 0.408i·6-s − 0.512i·7-s + 0.353i·8-s + 0.333·9-s − 0.316·10-s − 1.59i·11-s + 0.288·12-s − 0.362·14-s + 0.258i·15-s + 0.250·16-s + 0.496·17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8968164779\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8968164779\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 1.35iT - 7T^{2} \) |
| 11 | \( 1 + 5.29iT - 11T^{2} \) |
| 17 | \( 1 - 2.04T + 17T^{2} \) |
| 19 | \( 1 + 3.24iT - 19T^{2} \) |
| 23 | \( 1 + 1.33T + 23T^{2} \) |
| 29 | \( 1 + 5.13T + 29T^{2} \) |
| 31 | \( 1 + 2.80iT - 31T^{2} \) |
| 37 | \( 1 + 2.24iT - 37T^{2} \) |
| 41 | \( 1 + 10.6iT - 41T^{2} \) |
| 43 | \( 1 + 4.41T + 43T^{2} \) |
| 47 | \( 1 - 4.82iT - 47T^{2} \) |
| 53 | \( 1 - 14.2T + 53T^{2} \) |
| 59 | \( 1 - 13.8iT - 59T^{2} \) |
| 61 | \( 1 + 3.62T + 61T^{2} \) |
| 67 | \( 1 + 9.03iT - 67T^{2} \) |
| 71 | \( 1 + 11.7iT - 71T^{2} \) |
| 73 | \( 1 + 12.1iT - 73T^{2} \) |
| 79 | \( 1 - 0.259T + 79T^{2} \) |
| 83 | \( 1 + 3.02iT - 83T^{2} \) |
| 89 | \( 1 + 5.47iT - 89T^{2} \) |
| 97 | \( 1 - 5.85iT - 97T^{2} \) |
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
−7.78320627855620679244979802183, −7.17536559689073985508835423833, −6.01499143799161040204769290138, −5.64372247852261088307276215035, −4.77608244858398431089476539800, −3.92377408544048536593085586399, −3.32603190612409698154885624543, −2.17955412239119229068525332489, −0.995255823029770384440901979685, −0.31511429372794065955509428860,
1.43912050338735408733962110980, 2.43002090579720211344102340736, 3.65045148389312122381617627504, 4.39815782106215865604182098513, 5.27515675883168009799221302262, 5.72327487765251187106968308925, 6.67623890961576579914853671862, 7.06645548350022802401735590272, 7.83021856151047039731492499715, 8.485356456950799193952828308104