L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (−0.445 + 2.19i)5-s + (0.499 − 0.866i)6-s + 4.67i·7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (−0.710 − 2.12i)10-s − 3.96·11-s + 0.999i·12-s + (0.698 + 0.403i)13-s + (−2.33 − 4.04i)14-s + (−0.710 − 2.12i)15-s + (−0.5 − 0.866i)16-s + (4.01 − 2.31i)17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.199 + 0.979i)5-s + (0.204 − 0.353i)6-s + 1.76i·7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.224 − 0.670i)10-s − 1.19·11-s + 0.288i·12-s + (0.193 + 0.111i)13-s + (−0.624 − 1.08i)14-s + (−0.183 − 0.547i)15-s + (−0.125 − 0.216i)16-s + (0.974 − 0.562i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0873263 - 0.494181i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0873263 - 0.494181i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (0.445 - 2.19i)T \) |
| 19 | \( 1 + (-3.01 - 3.15i)T \) |
good | 7 | \( 1 - 4.67iT - 7T^{2} \) |
| 11 | \( 1 + 3.96T + 11T^{2} \) |
| 13 | \( 1 + (-0.698 - 0.403i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-4.01 + 2.31i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (5.52 + 3.19i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.03 - 3.51i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 3.35T + 31T^{2} \) |
| 37 | \( 1 + 2.19iT - 37T^{2} \) |
| 41 | \( 1 + (2.02 + 3.51i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (6.36 - 3.67i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (8.12 + 4.69i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.34 + 0.778i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.94 - 6.83i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.20 + 3.81i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.42 - 3.70i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.08 - 1.87i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-10.2 + 5.91i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.67 + 8.09i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 9.92iT - 83T^{2} \) |
| 89 | \( 1 + (9.13 - 15.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (10.7 - 6.20i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.17199625635442225554043686298, −10.13177861661907169285281694735, −9.717273892620684086915094153693, −8.447746175344577317253767920998, −7.81841389728730829576690320046, −6.65765106934354850799506134285, −5.74832763868489669324811915273, −5.18952897069874807708642925710, −3.30009393588390727619051794219, −2.22706662608221637647831393893,
0.37730676226889435342019738786, 1.49714096434320371940017866081, 3.44164166226445083775426457609, 4.52511988335714659672084510399, 5.55848612564724192235843709560, 6.88991371228833436666768197322, 7.963109952198358945650159738521, 8.035546103685318611145684157443, 9.786057596374333244831004803955, 10.09238334472073020734884420814