L(s) = 1 | − 2-s + (1.5 − 0.866i)3-s + 4-s + (1 − 1.73i)5-s + (−1.5 + 0.866i)6-s − 8-s + (1.5 − 2.59i)9-s + (−1 + 1.73i)10-s + (−0.5 − 0.866i)11-s + (1.5 − 0.866i)12-s + (−3 − 5.19i)13-s − 3.46i·15-s + 16-s + (−2.5 + 4.33i)17-s + (−1.5 + 2.59i)18-s + (−3.5 − 6.06i)19-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.866 − 0.499i)3-s + 0.5·4-s + (0.447 − 0.774i)5-s + (−0.612 + 0.353i)6-s − 0.353·8-s + (0.5 − 0.866i)9-s + (−0.316 + 0.547i)10-s + (−0.150 − 0.261i)11-s + (0.433 − 0.249i)12-s + (−0.832 − 1.44i)13-s − 0.894i·15-s + 0.250·16-s + (−0.606 + 1.05i)17-s + (−0.353 + 0.612i)18-s + (−0.802 − 1.39i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.415 + 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.755465 - 1.17505i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.755465 - 1.17505i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-1.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3 + 5.19i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.5 - 4.33i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2 + 3.46i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 7T + 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 - 13T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 + (-8 + 13.8i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.5 - 4.33i)T + (-48.5 - 84.0i)T^{2} \) |
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
−9.727441805925686140107129515604, −8.866934552999432882079680967780, −8.329195165711868322342342496364, −7.60830750679636519801628642651, −6.61547509958852745718702516373, −5.65972631198221324511713976871, −4.44007009865832714635654359256, −2.99637338889730586992485892775, −2.08504426574558671157875234302, −0.72886438361820211365616066580,
2.04445571257970849994995682780, 2.60537293457361060008247803849, 3.98163625808823220186171275637, 4.97042087332050589331086009998, 6.54013398236485940008217571986, 6.97781512180445837306400454561, 8.088511044148036834990943646146, 8.769191358265987266659673664024, 9.781906937207063259766060784799, 10.01774042577781044569141894288