L(s) = 1 | + (1.13 + 1.96i)2-s + (0.5 − 0.866i)3-s + (−1.58 + 2.74i)4-s + (−2.07 − 3.59i)5-s + 2.27·6-s + (−1.69 − 2.03i)7-s − 2.64·8-s + (−0.499 − 0.866i)9-s + (4.71 − 8.17i)10-s + (1.59 − 2.76i)11-s + (1.58 + 2.74i)12-s + 0.175·13-s + (2.08 − 5.64i)14-s − 4.15·15-s + (0.156 + 0.270i)16-s + (0.663 − 1.14i)17-s + ⋯ |
L(s) = 1 | + (0.803 + 1.39i)2-s + (0.288 − 0.499i)3-s + (−0.791 + 1.37i)4-s + (−0.928 − 1.60i)5-s + 0.927·6-s + (−0.639 − 0.768i)7-s − 0.936·8-s + (−0.166 − 0.288i)9-s + (1.49 − 2.58i)10-s + (0.480 − 0.832i)11-s + (0.456 + 0.791i)12-s + 0.0487·13-s + (0.556 − 1.50i)14-s − 1.07·15-s + (0.0390 + 0.0676i)16-s + (0.160 − 0.278i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.67621 - 0.354493i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.67621 - 0.354493i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (1.69 + 2.03i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-1.13 - 1.96i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (2.07 + 3.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.59 + 2.76i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 0.175T + 13T^{2} \) |
| 17 | \( 1 + (-0.663 + 1.14i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0252 + 0.0436i)T + (-9.5 + 16.4i)T^{2} \) |
| 29 | \( 1 - 1.10T + 29T^{2} \) |
| 31 | \( 1 + (-4.35 + 7.54i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.88 + 3.25i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 4.84T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + (-4.47 - 7.75i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.98 + 8.63i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.19 + 5.53i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.61 - 6.26i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.01 - 5.21i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6.71T + 71T^{2} \) |
| 73 | \( 1 + (-2.30 + 3.99i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.74 - 13.4i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 7.21T + 83T^{2} \) |
| 89 | \( 1 + (-0.970 - 1.68i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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
−11.29123271838147190855394576787, −9.609712346021600688375234829901, −8.627259998433891908405580413631, −8.028603684231789253958443974596, −7.26819887392800533794004781951, −6.30683705125668241752220823999, −5.31697640186767448771558463839, −4.25695546895641842709474458823, −3.58743315325005279377968649464, −0.814090694750727083414636215371,
2.27501025536518949375066765892, 3.16416138126449895106517805066, 3.77459631805805611650195193353, 4.86361504532602463103856251409, 6.31288381194569976460868929205, 7.25862011868388112799848448388, 8.586262837046732801763725509156, 9.840737815364508812920517492763, 10.32231176298318989741328681899, 11.09630965071401166465835123607