L(s) = 1 | + 1.73i·2-s − 0.999·4-s + (1.5 − 2.59i)5-s + (2 + 1.73i)7-s + 1.73i·8-s + (4.5 + 2.59i)10-s + (−1.5 + 0.866i)11-s + (1.5 − 0.866i)13-s + (−2.99 + 3.46i)14-s − 5·16-s + (1.5 − 2.59i)17-s + (−4.5 + 2.59i)19-s + (−1.49 + 2.59i)20-s + (−1.49 − 2.59i)22-s + (−4.5 − 2.59i)23-s + ⋯ |
L(s) = 1 | + 1.22i·2-s − 0.499·4-s + (0.670 − 1.16i)5-s + (0.755 + 0.654i)7-s + 0.612i·8-s + (1.42 + 0.821i)10-s + (−0.452 + 0.261i)11-s + (0.416 − 0.240i)13-s + (−0.801 + 0.925i)14-s − 1.25·16-s + (0.363 − 0.630i)17-s + (−1.03 + 0.596i)19-s + (−0.335 + 0.580i)20-s + (−0.319 − 0.553i)22-s + (−0.938 − 0.541i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.09372 + 0.860197i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.09372 + 0.860197i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
good | 2 | \( 1 - 1.73iT - 2T^{2} \) |
| 5 | \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 - 0.866i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.5 + 0.866i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.5 - 2.59i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.5 + 2.59i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.5 - 2.59i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 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 | \( 1 + (7.5 + 4.33i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13.8iT - 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 3.46iT - 71T^{2} \) |
| 73 | \( 1 + (4.5 + 2.59i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + (7.5 - 12.9i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.5 - 2.59i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.5 - 0.866i)T + (48.5 + 84.0i)T^{2} \) |
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
−12.86706278285562702997062033078, −12.04191054325712859493301479954, −10.76065097039486489265060144778, −9.396968915543597441072449387229, −8.444276237637559525034583011494, −7.88773791311141527155313763300, −6.33134503448031498332437570336, −5.46118461214875833774081558645, −4.68673761222492747515077923309, −2.05497299292172316669943338762,
1.75705901659726654071396250373, 3.00861810588390214752082583317, 4.29954279178598716552846928541, 6.10379108144574420531505957857, 7.12425691836572134446665467770, 8.487508167254132979072366112366, 10.00279037089535379135670466250, 10.53119838694185963234949928565, 11.13167810783007110224673052767, 12.11376097525866690997451635786