L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + i·5-s + (−0.866 + 0.499i)6-s + (−1.73 + i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)10-s + (5.59 + 3.23i)11-s + 0.999·12-s + (1 − 3.46i)13-s + 1.99·14-s + (0.866 + 0.5i)15-s + (−0.5 + 0.866i)16-s + (2 + 3.46i)17-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s + 0.447i·5-s + (−0.353 + 0.204i)6-s + (−0.654 + 0.377i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.158 − 0.273i)10-s + (1.68 + 0.974i)11-s + 0.288·12-s + (0.277 − 0.960i)13-s + 0.534·14-s + (0.223 + 0.129i)15-s + (−0.125 + 0.216i)16-s + (0.485 + 0.840i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.15280 - 0.155531i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.15280 - 0.155531i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (-1 + 3.46i)T \) |
good | 7 | \( 1 + (1.73 - i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.59 - 3.23i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.46 + 3.73i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.86 - 3.23i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.133 - 0.232i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 1.73iT - 31T^{2} \) |
| 37 | \( 1 + (-7.96 - 4.59i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.73 - i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.96 + 10.3i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 3.53iT - 47T^{2} \) |
| 53 | \( 1 - 0.928T + 53T^{2} \) |
| 59 | \( 1 + (-7.33 + 4.23i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.19 - 9i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (9.92 + 5.73i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (10.7 - 6.19i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 - 8.92iT - 83T^{2} \) |
| 89 | \( 1 + (0.464 + 0.267i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.464 + 0.267i)T + (48.5 - 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.46402647924584353568484137663, −10.06633265487803078178147423637, −9.569670128619518044723870423904, −8.626997887928937813892154417959, −7.52100024155861497188637053801, −6.79082252273857290333450803593, −5.77407397996638689192526797988, −3.85457115966113509359783228067, −2.87138954784061684477933101437, −1.37082138207289870932682328220,
1.17735732094833667893933524257, 3.28280422320829006610625162403, 4.31498322252172755025692376795, 5.78617412331234509461318941352, 6.63623566718143719176609460135, 7.75357349541346647192320341784, 8.847346073073651922932644429919, 9.412425011472397068456333371022, 10.05719180685212250643458355281, 11.42391371656786079508589163953