L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (−1 − 2i)5-s + (−0.499 + 0.866i)6-s + (3.46 + 2i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−1.86 − 1.23i)10-s + (−2.5 − 4.33i)11-s + 0.999i·12-s + (3.46 + i)13-s + 3.99·14-s + (1.86 + 1.23i)15-s + (−0.5 − 0.866i)16-s + (−1.73 − i)17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.447 − 0.894i)5-s + (−0.204 + 0.353i)6-s + (1.30 + 0.755i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.590 − 0.389i)10-s + (−0.753 − 1.30i)11-s + 0.288i·12-s + (0.960 + 0.277i)13-s + 1.06·14-s + (0.481 + 0.318i)15-s + (−0.125 − 0.216i)16-s + (−0.420 − 0.242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.42909 - 0.895942i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.42909 - 0.895942i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1 + 2i)T \) |
| 13 | \( 1 + (-3.46 - i)T \) |
good | 7 | \( 1 + (-3.46 - 2i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.73 + i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4 + 6.92i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.59 + 1.5i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.5 - 6.06i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 + (-2.59 + 1.5i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.866 - 0.5i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 7iT - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + (2.5 - 4.33i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2 - 3.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.46 - 2i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4 - 6.92i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (10.3 + 6i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.16086348442698216737435076929, −10.89519117396369120162221683989, −9.034494969992397769736321753205, −8.692848138308559994320676220530, −7.41875507973036121836403094390, −5.85876874669041225382671739283, −5.16407868492461975782949194343, −4.45635579112063896931716342066, −2.97265521241829789945219663182, −1.12932982314036409475220127700,
1.86470832754220607218138244917, 3.63170623858383720726006229690, 4.63200287293235525627452814158, 5.66652588577871367726047655829, 6.83217662294791915136496288166, 7.67679244320115573065542070755, 8.074438177838774821909957434788, 10.03858670832165308820386907410, 10.81247734460394356865631440077, 11.46079473881670930156691853184