| L(s) = 1 | + (−0.951 + 0.309i)2-s + (−0.587 − 0.809i)3-s + (0.809 − 0.587i)4-s + (−2.06 + 0.847i)5-s + (0.809 + 0.587i)6-s − 4.07i·7-s + (−0.587 + 0.809i)8-s + (−0.309 + 0.951i)9-s + (1.70 − 1.44i)10-s + (−1.01 − 3.12i)11-s + (−0.951 − 0.309i)12-s + (−5.15 − 1.67i)13-s + (1.26 + 3.87i)14-s + (1.90 + 1.17i)15-s + (0.309 − 0.951i)16-s + (2.03 − 2.79i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 + 0.218i)2-s + (−0.339 − 0.467i)3-s + (0.404 − 0.293i)4-s + (−0.925 + 0.379i)5-s + (0.330 + 0.239i)6-s − 1.54i·7-s + (−0.207 + 0.286i)8-s + (−0.103 + 0.317i)9-s + (0.539 − 0.457i)10-s + (−0.305 − 0.941i)11-s + (−0.274 − 0.0892i)12-s + (−1.43 − 0.464i)13-s + (0.336 + 1.03i)14-s + (0.491 + 0.303i)15-s + (0.0772 − 0.237i)16-s + (0.493 − 0.678i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.404 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.404 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.246931 - 0.379044i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.246931 - 0.379044i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 3 | \( 1 + (0.587 + 0.809i)T \) |
| 5 | \( 1 + (2.06 - 0.847i)T \) |
| good | 7 | \( 1 + 4.07iT - 7T^{2} \) |
| 11 | \( 1 + (1.01 + 3.12i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (5.15 + 1.67i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.03 + 2.79i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.32 - 0.961i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (0.581 - 0.188i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (3.78 - 2.74i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-6.71 - 4.88i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.28 - 0.741i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.905 + 2.78i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 8.64iT - 43T^{2} \) |
| 47 | \( 1 + (4.06 + 5.59i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-3.99 - 5.49i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-4.38 + 13.4i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (3.88 + 11.9i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-6.39 + 8.79i)T + (-20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-7.33 + 5.32i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (8.65 - 2.81i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (3.31 - 2.40i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-4.19 + 5.76i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (1.48 + 4.56i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-8.67 - 11.9i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−12.53226102659801423458019422582, −11.48514607239862653845922728515, −10.69348234935064364075143886178, −9.830802228470656905844632670004, −8.062529294984316353423098736177, −7.52560620207427482449308985553, −6.66718502674390051944753895411, −4.96820418358940770032079564638, −3.21801842446514913325743962387, −0.56250169283673559377675110514,
2.49655488322092396469502388147, 4.37543552111195535176490240179, 5.59586891974056645186885715717, 7.23432075200131771854728269474, 8.293473157698899063349116644743, 9.333967678576151348287557948447, 10.08949606559614078859180677483, 11.60594822408303122264317717431, 11.99800713106492683052878727588, 12.78442801576331476583681166499
