| L(s) = 1 | + (−0.939 − 0.342i)2-s + (−0.173 − 0.984i)3-s + (0.766 + 0.642i)4-s + (−0.766 + 0.642i)5-s + (−0.173 + 0.984i)6-s + (−0.439 + 0.761i)7-s + (−0.500 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (0.939 − 0.342i)10-s + (−0.266 − 0.460i)11-s + (0.5 − 0.866i)12-s + (0.141 − 0.802i)13-s + (0.673 − 0.565i)14-s + (0.766 + 0.642i)15-s + (0.173 + 0.984i)16-s + (−5.08 − 1.85i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (−0.100 − 0.568i)3-s + (0.383 + 0.321i)4-s + (−0.342 + 0.287i)5-s + (−0.0708 + 0.402i)6-s + (−0.166 + 0.287i)7-s + (−0.176 − 0.306i)8-s + (−0.313 + 0.114i)9-s + (0.297 − 0.108i)10-s + (−0.0802 − 0.138i)11-s + (0.144 − 0.249i)12-s + (0.0392 − 0.222i)13-s + (0.180 − 0.151i)14-s + (0.197 + 0.165i)15-s + (0.0434 + 0.246i)16-s + (−1.23 − 0.448i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.838 - 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.838 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00526586 + 0.0177409i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00526586 + 0.0177409i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 + 0.342i)T \) |
| 3 | \( 1 + (0.173 + 0.984i)T \) |
| 5 | \( 1 + (0.766 - 0.642i)T \) |
| 19 | \( 1 + (3.37 - 2.75i)T \) |
| good | 7 | \( 1 + (0.439 - 0.761i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.266 + 0.460i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.141 + 0.802i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (5.08 + 1.85i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (4.02 + 3.37i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (5.14 - 1.87i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (1.29 - 2.24i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2.53T + 37T^{2} \) |
| 41 | \( 1 + (-0.354 - 2.01i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (1.31 - 1.10i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (8.69 - 3.16i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (2.17 + 1.82i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (0.152 + 0.0555i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.08 - 0.907i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (10.9 - 4.00i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (5.88 - 4.93i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (1.87 + 10.6i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.843 - 4.78i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-7.02 + 12.1i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.699 - 3.96i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-6.88 - 2.50i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.06198445135653152671119238864, −10.34298027672830302129961344237, −9.236296137200767209844387931630, −8.438734480205872910128505700983, −7.65645613991352446693435074316, −6.70171692755085262373972478269, −5.93466077707259585822006244620, −4.39885244447945918953071918774, −3.03233851141583655801700967586, −1.90158095164971617548346184282,
0.01228722338216915245269650829, 2.07660420381021449072933724231, 3.75520005866464573697450137507, 4.67150883640759677114843653405, 5.90156774036923896784972100813, 6.83694257825963537865494742042, 7.83941955310342814728318582938, 8.741994572657832883763409581994, 9.419599043452763744868509716704, 10.31387159166454076721633359108
