| L(s) = 1 | + (−0.972 + 0.816i)3-s + (−1.43 − 0.524i)5-s + (1.82 + 0.665i)7-s + (−0.240 + 1.36i)9-s + (0.220 − 0.381i)11-s + (−0.367 − 2.08i)13-s + (1.82 − 0.665i)15-s + (−0.664 + 3.76i)17-s + (−4.55 + 3.82i)19-s + (−2.32 + 0.845i)21-s + (2.85 + 4.94i)23-s + (−2.03 − 1.70i)25-s + (−2.78 − 4.82i)27-s + (−1.36 + 2.36i)29-s − 9.87·31-s + ⋯ |
| L(s) = 1 | + (−0.561 + 0.471i)3-s + (−0.643 − 0.234i)5-s + (0.691 + 0.251i)7-s + (−0.0802 + 0.455i)9-s + (0.0664 − 0.115i)11-s + (−0.101 − 0.578i)13-s + (0.472 − 0.171i)15-s + (−0.161 + 0.913i)17-s + (−1.04 + 0.876i)19-s + (−0.506 + 0.184i)21-s + (0.595 + 1.03i)23-s + (−0.406 − 0.341i)25-s + (−0.536 − 0.928i)27-s + (−0.253 + 0.439i)29-s − 1.77·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.705 - 0.708i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.705 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.259355 + 0.624108i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.259355 + 0.624108i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 + (-2.18 - 5.67i)T \) |
| good | 3 | \( 1 + (0.972 - 0.816i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (1.43 + 0.524i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-1.82 - 0.665i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (-0.220 + 0.381i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.367 + 2.08i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.664 - 3.76i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (4.55 - 3.82i)T + (3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (-2.85 - 4.94i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.36 - 2.36i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 9.87T + 31T^{2} \) |
| 41 | \( 1 + (-1.80 - 10.2i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + 3.84T + 43T^{2} \) |
| 47 | \( 1 + (-3.84 - 6.65i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.62 - 0.591i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.93 + 0.702i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (1.89 + 10.7i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (1.57 + 0.572i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-7.70 + 6.46i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + 9.88T + 73T^{2} \) |
| 79 | \( 1 + (-12.5 - 4.56i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (0.617 - 3.50i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-11.1 + 4.05i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-0.296 - 0.513i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.02871976771130169841418650258, −10.39250584983418428586477873811, −9.290602109093844591852987998100, −8.106619712060588957660624456321, −7.85142047742986120255303207121, −6.29651661440702266487837074701, −5.38416387969227217380541951448, −4.56993958142065837052366236963, −3.55320541757081223603636114500, −1.80300019237093673599269532904,
0.39929017654160560364234508128, 2.14580701342357175855241386572, 3.75449822519437113638540764232, 4.71112392996853615274555221529, 5.83962227626755100158590310393, 7.06393765014801464357447915930, 7.27795327190393839112986634459, 8.659002468426094498941750519388, 9.321398305346677368940234760839, 10.76475513772125954999981393922
