| L(s) = 1 | + (−1.78 + 1.15i)3-s + (−0.356 − 0.781i)5-s + (−3.61 + 1.06i)7-s + (0.634 − 1.38i)9-s + (−1.12 − 1.29i)11-s + (−5.14 − 1.51i)13-s + (1.53 + 0.987i)15-s + (−0.432 + 3.01i)17-s + (0.0142 + 0.0991i)19-s + (5.24 − 6.05i)21-s + (−2.50 + 4.09i)23-s + (2.79 − 3.22i)25-s + (−0.445 − 3.10i)27-s + (−0.849 + 5.90i)29-s + (6.18 + 3.97i)31-s + ⋯ |
| L(s) = 1 | + (−1.03 + 0.664i)3-s + (−0.159 − 0.349i)5-s + (−1.36 + 0.401i)7-s + (0.211 − 0.463i)9-s + (−0.339 − 0.391i)11-s + (−1.42 − 0.419i)13-s + (0.396 + 0.255i)15-s + (−0.104 + 0.730i)17-s + (0.00326 + 0.0227i)19-s + (1.14 − 1.32i)21-s + (−0.521 + 0.852i)23-s + (0.558 − 0.644i)25-s + (−0.0858 − 0.596i)27-s + (−0.157 + 1.09i)29-s + (1.11 + 0.713i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0102i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0102i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.000827825 - 0.160891i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.000827825 - 0.160891i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (2.50 - 4.09i)T \) |
| good | 3 | \( 1 + (1.78 - 1.15i)T + (1.24 - 2.72i)T^{2} \) |
| 5 | \( 1 + (0.356 + 0.781i)T + (-3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (3.61 - 1.06i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (1.12 + 1.29i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (5.14 + 1.51i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.432 - 3.01i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.0142 - 0.0991i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (0.849 - 5.90i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-6.18 - 3.97i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (0.00609 - 0.0133i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-2.66 - 5.83i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-5.30 + 3.41i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 6.93T + 47T^{2} \) |
| 53 | \( 1 + (4.36 - 1.28i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (9.27 + 2.72i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (11.1 + 7.15i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-1.87 + 2.16i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-8.91 + 10.2i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.57 - 10.9i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (9.35 + 2.74i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (4.75 - 10.4i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (14.2 - 9.16i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (5.55 + 12.1i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−12.60827457770076600967921188771, −12.34884634366929278351446949938, −11.05527637924968940388142068336, −10.14864825459818893609013975557, −9.457858528172767036910614027697, −8.081084248385271868686295933768, −6.58785616932862026184668145956, −5.63064991630050999691599765488, −4.64721900564830721396846008449, −3.04377763532304098075218258678,
0.15266850604348118393862471727, 2.74148895031069136467519163449, 4.54838379636506773050018879710, 5.99871222010162335160939592053, 6.85652887734245294817680549547, 7.53325467945431358543933813349, 9.420306447905198565966984630235, 10.15966174202745132209319663465, 11.31987570970069943494528936881, 12.23351055572399022238254744568
