| L(s) = 1 | + (−1.98 + 0.260i)2-s + (1.49 + 2.60i)3-s + (3.86 − 1.03i)4-s + (1.48 + 8.44i)5-s + (−3.63 − 4.77i)6-s + (6.53 + 5.48i)7-s + (−7.39 + 3.05i)8-s + (−4.54 + 7.76i)9-s + (−5.15 − 16.3i)10-s + (−0.347 + 1.96i)11-s + (8.45 + 8.51i)12-s + (5.67 − 15.5i)13-s + (−14.3 − 9.16i)14-s + (−19.7 + 16.4i)15-s + (13.8 − 7.97i)16-s + (16.0 − 9.26i)17-s + ⋯ |
| L(s) = 1 | + (−0.991 + 0.130i)2-s + (0.497 + 0.867i)3-s + (0.966 − 0.257i)4-s + (0.297 + 1.68i)5-s + (−0.606 − 0.795i)6-s + (0.933 + 0.782i)7-s + (−0.924 + 0.381i)8-s + (−0.504 + 0.863i)9-s + (−0.515 − 1.63i)10-s + (−0.0315 + 0.178i)11-s + (0.704 + 0.709i)12-s + (0.436 − 1.19i)13-s + (−1.02 − 0.654i)14-s + (−1.31 + 1.09i)15-s + (0.866 − 0.498i)16-s + (0.943 − 0.544i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.623 - 0.781i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.576486 + 1.19802i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.576486 + 1.19802i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.98 - 0.260i)T \) |
| 3 | \( 1 + (-1.49 - 2.60i)T \) |
| good | 5 | \( 1 + (-1.48 - 8.44i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-6.53 - 5.48i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (0.347 - 1.96i)T + (-113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-5.67 + 15.5i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-16.0 + 9.26i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (4.57 + 2.64i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (11.2 + 13.3i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-8.70 + 3.17i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-37.8 + 31.7i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (43.2 - 24.9i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (27.2 - 74.8i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-16.2 - 2.87i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (-40.1 + 47.8i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + 15.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + (4.53 + 25.7i)T + (-3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-40.8 + 48.6i)T + (-646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (0.672 - 1.84i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-63.7 + 36.8i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (21.4 - 37.1i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-86.3 + 31.4i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-23.8 + 8.69i)T + (5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (68.4 + 39.5i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (13.3 - 75.8i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−11.88788995638534047789698489262, −11.08543009742135459304780656932, −10.29764215399002644559999825043, −9.764868137167914430660137980104, −8.358107984301901174868619530306, −7.81841843093365961524343653162, −6.40996535318859309071743259509, −5.31783660648373461962453969079, −3.18625668857509406653875619287, −2.32806449239267176322570508562,
1.05640588241440376900599978702, 1.76576561058496525131883941736, 3.98034601124524566565348066913, 5.65111260588630224082004384074, 7.01069299218758165350674582837, 8.102262653936483445385234731108, 8.592107538222170625454185484864, 9.440441126558063412554589103713, 10.70161800527007724451755338129, 12.01842007375296516292625104991
