| L(s) = 1 | + (1.55 − 0.273i)2-s + (−2.81 − 1.03i)3-s + (−1.42 + 0.518i)4-s + (2.85 + 3.40i)5-s + (−4.65 − 0.832i)6-s + (−0.447 − 0.775i)7-s + (−7.52 + 4.34i)8-s + (6.86 + 5.81i)9-s + (5.36 + 4.50i)10-s + 17.0i·11-s + (4.54 + 0.0112i)12-s + (13.4 + 11.2i)13-s + (−0.906 − 1.08i)14-s + (−4.52 − 12.5i)15-s + (−5.84 + 4.90i)16-s + (−8.14 − 9.71i)17-s + ⋯ |
| L(s) = 1 | + (0.776 − 0.136i)2-s + (−0.938 − 0.344i)3-s + (−0.356 + 0.129i)4-s + (0.571 + 0.680i)5-s + (−0.775 − 0.138i)6-s + (−0.0639 − 0.110i)7-s + (−0.941 + 0.543i)8-s + (0.762 + 0.646i)9-s + (0.536 + 0.450i)10-s + 1.55i·11-s + (0.379 + 0.000934i)12-s + (1.03 + 0.867i)13-s + (−0.0647 − 0.0771i)14-s + (−0.301 − 0.835i)15-s + (−0.365 + 0.306i)16-s + (−0.479 − 0.571i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.984023 + 0.812966i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.984023 + 0.812966i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.81 + 1.03i)T \) |
| 19 | \( 1 + (16.4 - 9.57i)T \) |
| good | 2 | \( 1 + (-1.55 + 0.273i)T + (3.75 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-2.85 - 3.40i)T + (-4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (0.447 + 0.775i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 - 17.0iT - 121T^{2} \) |
| 13 | \( 1 + (-13.4 - 11.2i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (8.14 + 9.71i)T + (-50.1 + 284. i)T^{2} \) |
| 23 | \( 1 + (1.97 + 5.41i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (-15.6 - 42.9i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 - 1.22T + 961T^{2} \) |
| 37 | \( 1 + 40.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-41.2 + 7.27i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (11.6 + 4.24i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (12.6 + 34.8i)T + (-1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + (-36.6 - 6.46i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (20.1 - 55.2i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (34.0 + 28.5i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (7.14 - 40.4i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-121. + 21.4i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (-65.5 - 23.8i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (109. - 92.0i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-74.7 + 43.1i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-7.26 - 19.9i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (-23.6 - 133. i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−12.66885234260490831349248510503, −12.03877431099866352284819842658, −10.90811887795737327137531728868, −10.01615862431937801799834644276, −8.740979825305460183295089543938, −7.02941577325920280604170303585, −6.33495550381563515095033115311, −5.06652099291922360596101400163, −4.06119929147255265079723739151, −2.10878179028028388093142292162,
0.70899634418683634484954241417, 3.57737060068092021731993067538, 4.76559262296998578621315441424, 5.90678820363781993440442870477, 6.15975270933414235592175611052, 8.442376322880049993440580361434, 9.263659758997763153310451264758, 10.49462297500932233387820500661, 11.34772802092779373045503193609, 12.62561506977853518519236878801
