L(s) = 1 | + (−0.809 − 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (−2.65 + 1.92i)5-s + (0.809 − 0.587i)6-s + (−0.309 − 0.951i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + 3.28·10-s + (2.53 + 2.13i)11-s − 12-s + (−2.52 − 1.83i)13-s + (−0.309 + 0.951i)14-s + (−1.01 − 3.11i)15-s + (−0.809 + 0.587i)16-s + (−3.18 + 2.31i)17-s + ⋯ |
L(s) = 1 | + (−0.572 − 0.415i)2-s + (−0.178 + 0.549i)3-s + (0.154 + 0.475i)4-s + (−1.18 + 0.862i)5-s + (0.330 − 0.239i)6-s + (−0.116 − 0.359i)7-s + (0.109 − 0.336i)8-s + (−0.269 − 0.195i)9-s + 1.03·10-s + (0.764 + 0.644i)11-s − 0.288·12-s + (−0.699 − 0.508i)13-s + (−0.0825 + 0.254i)14-s + (−0.261 − 0.805i)15-s + (−0.202 + 0.146i)16-s + (−0.773 + 0.562i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 + 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0213768 - 0.0716251i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0213768 - 0.0716251i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 3 | \( 1 + (0.309 - 0.951i)T \) |
| 7 | \( 1 + (0.309 + 0.951i)T \) |
| 11 | \( 1 + (-2.53 - 2.13i)T \) |
good | 5 | \( 1 + (2.65 - 1.92i)T + (1.54 - 4.75i)T^{2} \) |
| 13 | \( 1 + (2.52 + 1.83i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (3.18 - 2.31i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.74 + 5.36i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 4.33T + 23T^{2} \) |
| 29 | \( 1 + (3.11 + 9.57i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (8.56 + 6.22i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.249 - 0.768i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.864 + 2.66i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 8.17T + 43T^{2} \) |
| 47 | \( 1 + (3.52 - 10.8i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.41 - 1.02i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (2.68 + 8.26i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (7.12 - 5.17i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + (2.85 - 2.07i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.196 - 0.603i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-9.41 - 6.83i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (12.5 - 9.08i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 - 7.99T + 89T^{2} \) |
| 97 | \( 1 + (-11.0 - 8.02i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.85663840921287924779442700458, −9.816371941044134259730682922688, −9.158926365628111567273358016072, −7.77940342396964083196767827866, −7.32979453111625836187151385672, −6.18760227713615111442494000619, −4.40224968919230912356953890683, −3.79824358600560736851313268079, −2.48891271095644387884175389610, −0.05660711863752189011835790835,
1.60447331561164306161180348848, 3.59042764927761029864947039138, 4.86799227194430527831477217195, 5.92121970097444575966143649939, 7.08289684019498216557720644112, 7.72197404480043650722299113349, 8.809502566638497680558093313657, 9.115588827917899703764160500085, 10.58090330045012342895167258527, 11.64489441541460163168464821999