| L(s) = 1 | + (−0.618 − 1.90i)2-s + (4.17 − 3.03i)3-s + (−3.23 + 2.35i)4-s + (−8.34 − 6.06i)6-s + 3.36·7-s + (6.47 + 4.70i)8-s + (−0.124 + 0.383i)9-s + (−4.26 − 13.1i)11-s + (−6.37 + 19.6i)12-s + (28.1 − 86.7i)13-s + (−2.07 − 6.39i)14-s + (4.94 − 15.2i)16-s + (−44.3 − 32.2i)17-s + 0.806·18-s + (−49.0 − 35.6i)19-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (0.802 − 0.583i)3-s + (−0.404 + 0.293i)4-s + (−0.567 − 0.412i)6-s + 0.181·7-s + (0.286 + 0.207i)8-s + (−0.00461 + 0.0142i)9-s + (−0.116 − 0.360i)11-s + (−0.153 + 0.471i)12-s + (0.601 − 1.85i)13-s + (−0.0396 − 0.122i)14-s + (0.0772 − 0.237i)16-s + (−0.632 − 0.459i)17-s + 0.0105·18-s + (−0.592 − 0.430i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 + 0.547i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.483830 - 1.62373i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.483830 - 1.62373i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.618 + 1.90i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (-4.17 + 3.03i)T + (8.34 - 25.6i)T^{2} \) |
| 7 | \( 1 - 3.36T + 343T^{2} \) |
| 11 | \( 1 + (4.26 + 13.1i)T + (-1.07e3 + 782. i)T^{2} \) |
| 13 | \( 1 + (-28.1 + 86.7i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (44.3 + 32.2i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (49.0 + 35.6i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + (41.3 + 127. i)T + (-9.84e3 + 7.15e3i)T^{2} \) |
| 29 | \( 1 + (-125. + 90.8i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (77.5 + 56.3i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (4.12 - 12.6i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (47.9 - 147. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 - 471.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (145. - 105. i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (65.2 - 47.4i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (69.6 - 214. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-90.5 - 278. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + (814. + 591. i)T + (9.29e4 + 2.86e5i)T^{2} \) |
| 71 | \( 1 + (-791. + 575. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (249. + 769. i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-97.7 + 70.9i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-935. - 679. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + (-399. - 1.22e3i)T + (-5.70e5 + 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-1.32e3 + 963. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−11.00685304854704425937272519825, −10.49853912880352244068452204033, −9.130846387366896467617730469794, −8.269996337541394342146295307864, −7.72365656736590857277037428462, −6.19841366090155927774243778891, −4.74154166018124733874135339620, −3.18386039484593507762290045643, −2.29922538066669780121225329420, −0.65784281463471386596698839175,
1.85447032050923050413642730163, 3.72001115674016690901277046138, 4.55690487986235160866259051011, 6.10589223789264908146097166326, 7.06055636857564484734364668075, 8.338713523220947616682663902362, 8.998646210691774631558383460469, 9.725000048490753206849114197865, 10.86107940395757523006122373710, 11.95989718205363985966199469038
