L(s) = 1 | + (1.5 − 0.866i)3-s + (−1.41 + 2.44i)5-s + (1.5 − 2.59i)9-s + (−4.5 + 2.59i)11-s + (4.24 + 2.44i)13-s + 4.89i·15-s − 1.73i·17-s + 3·19-s + (−4.24 + 7.34i)23-s + (−1.49 − 2.59i)25-s − 5.19i·27-s + (2.82 + 4.89i)29-s + (−4.5 + 7.79i)33-s + 4.89i·37-s + 8.48·39-s + ⋯ |
L(s) = 1 | + (0.866 − 0.499i)3-s + (−0.632 + 1.09i)5-s + (0.5 − 0.866i)9-s + (−1.35 + 0.783i)11-s + (1.17 + 0.679i)13-s + 1.26i·15-s − 0.420i·17-s + 0.688·19-s + (−0.884 + 1.53i)23-s + (−0.299 − 0.519i)25-s − 0.999i·27-s + (0.525 + 0.909i)29-s + (−0.783 + 1.35i)33-s + 0.805i·37-s + 1.35·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.710856895\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.710856895\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 + 0.866i)T \) |
good | 5 | \( 1 + (1.41 - 2.44i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (4.5 - 2.59i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.24 - 2.44i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 1.73iT - 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 + (4.24 - 7.34i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.82 - 4.89i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.89iT - 37T^{2} \) |
| 41 | \( 1 + (-4.5 - 2.59i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.5 - 7.79i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.24 + 7.34i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 5.65T + 53T^{2} \) |
| 59 | \( 1 + (4.5 + 2.59i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (8.48 - 4.89i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.5 - 2.59i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 8.48T + 71T^{2} \) |
| 73 | \( 1 - 5T + 73T^{2} \) |
| 79 | \( 1 + (-12.7 + 7.34i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-9 + 5.19i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 13.8iT - 89T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.895411373608996373936431884990, −9.131965878033983264916792831443, −8.018799212911844435495240943791, −7.57751043712441284682486957019, −6.90928699014468036899918917940, −5.96600283097670784252017434950, −4.59146886092988384836453472701, −3.45755448290330726385707385480, −2.86997155378123363257134970115, −1.61543965145948125269807704952,
0.68582904099563324611671374407, 2.41018230680628188104764811609, 3.49864711531740078752660333351, 4.30151452355648023469411796665, 5.21152481912439668863685235386, 6.07805891935055811266064797121, 7.67582994830394062921998238378, 8.215787795973352379297209939172, 8.553059871460663541720125989232, 9.485642342591380137219278531823