| L(s) = 1 | + (1.08 − 0.905i)2-s + (0.0543 + 1.73i)3-s + (0.360 − 1.96i)4-s + (−2.37 + 2.70i)5-s + (1.62 + 1.83i)6-s + (0.599 − 0.0788i)7-s + (−1.38 − 2.46i)8-s + (−2.99 + 0.188i)9-s + (−0.128 + 5.09i)10-s + (−2.82 + 5.72i)11-s + (3.42 + 0.517i)12-s + (0.527 − 1.55i)13-s + (0.579 − 0.628i)14-s + (−4.81 − 3.96i)15-s + (−3.73 − 1.41i)16-s + (−2.48 + 2.48i)17-s + ⋯ |
| L(s) = 1 | + (0.768 − 0.640i)2-s + (0.0313 + 0.999i)3-s + (0.180 − 0.983i)4-s + (−1.06 + 1.21i)5-s + (0.663 + 0.747i)6-s + (0.226 − 0.0298i)7-s + (−0.491 − 0.871i)8-s + (−0.998 + 0.0626i)9-s + (−0.0406 + 1.61i)10-s + (−0.851 + 1.72i)11-s + (0.988 + 0.149i)12-s + (0.146 − 0.431i)13-s + (0.154 − 0.167i)14-s + (−1.24 − 1.02i)15-s + (−0.934 − 0.354i)16-s + (−0.602 + 0.602i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.309 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.748815 + 1.03159i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.748815 + 1.03159i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.08 + 0.905i)T \) |
| 3 | \( 1 + (-0.0543 - 1.73i)T \) |
| good | 5 | \( 1 + (2.37 - 2.70i)T + (-0.652 - 4.95i)T^{2} \) |
| 7 | \( 1 + (-0.599 + 0.0788i)T + (6.76 - 1.81i)T^{2} \) |
| 11 | \( 1 + (2.82 - 5.72i)T + (-6.69 - 8.72i)T^{2} \) |
| 13 | \( 1 + (-0.527 + 1.55i)T + (-10.3 - 7.91i)T^{2} \) |
| 17 | \( 1 + (2.48 - 2.48i)T - 17iT^{2} \) |
| 19 | \( 1 + (0.360 + 1.81i)T + (-17.5 + 7.27i)T^{2} \) |
| 23 | \( 1 + (0.431 - 3.28i)T + (-22.2 - 5.95i)T^{2} \) |
| 29 | \( 1 + (-7.89 - 0.517i)T + (28.7 + 3.78i)T^{2} \) |
| 31 | \( 1 + (-2.49 - 1.44i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.97 - 9.94i)T + (-34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (-0.516 + 3.92i)T + (-39.6 - 10.6i)T^{2} \) |
| 43 | \( 1 + (-6.04 - 2.98i)T + (26.1 + 34.1i)T^{2} \) |
| 47 | \( 1 + (-1.06 - 0.285i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.409 - 0.612i)T + (-20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (-6.59 + 7.52i)T + (-7.70 - 58.4i)T^{2} \) |
| 61 | \( 1 + (0.876 - 13.3i)T + (-60.4 - 7.96i)T^{2} \) |
| 67 | \( 1 + (-8.66 + 4.27i)T + (40.7 - 53.1i)T^{2} \) |
| 71 | \( 1 + (-0.0723 + 0.174i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (2.42 + 5.85i)T + (-51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (-1.21 + 4.53i)T + (-68.4 - 39.5i)T^{2} \) |
| 83 | \( 1 + (-7.35 + 6.44i)T + (10.8 - 82.2i)T^{2} \) |
| 89 | \( 1 + (4.84 + 2.00i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + (10.5 - 6.11i)T + (48.5 - 84.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.89171343861373779145436190394, −10.38196887164345191463609853462, −9.763816077426176360744104138761, −8.354907706744254162200285225012, −7.30180978874355211066306494635, −6.34672599667392226455919562648, −4.94480622991340159004716613189, −4.35864452801593588607001733437, −3.30947250693932112893822668048, −2.44885070175414085670523174872,
0.53992991782422059685679361207, 2.64040205702376978229814912412, 3.89683417169870598571338892399, 4.98179626410864236124681797633, 5.82799340235337294438225768680, 6.83661288170322771135919923136, 7.935348185541480678475147249084, 8.336092140940260395592230330699, 8.902504251150914131846547280012, 11.05566964855482572505475510251
