| L(s) = 1 | + (−0.326 + 1.37i)2-s + (2.56 + 1.55i)3-s + (−1.78 − 0.897i)4-s + (−0.00733 − 0.00546i)5-s + (−2.97 + 3.02i)6-s + (2.72 + 1.79i)7-s + (1.81 − 2.16i)8-s + (4.17 + 7.97i)9-s + (0.00991 − 0.00831i)10-s + (1.72 + 14.7i)11-s + (−3.19 − 5.07i)12-s + (−1.56 + 5.21i)13-s + (−3.35 + 3.16i)14-s + (−0.0103 − 0.0254i)15-s + (2.38 + 3.20i)16-s + (−6.21 + 1.09i)17-s + ⋯ |
| L(s) = 1 | + (−0.163 + 0.688i)2-s + (0.855 + 0.517i)3-s + (−0.446 − 0.224i)4-s + (−0.00146 − 0.00109i)5-s + (−0.495 + 0.504i)6-s + (0.389 + 0.256i)7-s + (0.227 − 0.270i)8-s + (0.464 + 0.885i)9-s + (0.000991 − 0.000831i)10-s + (0.157 + 1.34i)11-s + (−0.266 − 0.423i)12-s + (−0.120 + 0.401i)13-s + (−0.239 + 0.226i)14-s + (−0.000690 − 0.00169i)15-s + (0.149 + 0.200i)16-s + (−0.365 + 0.0644i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.245 - 0.969i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.05381 + 1.35446i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05381 + 1.35446i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.326 - 1.37i)T \) |
| 3 | \( 1 + (-2.56 - 1.55i)T \) |
| good | 5 | \( 1 + (0.00733 + 0.00546i)T + (7.17 + 23.9i)T^{2} \) |
| 7 | \( 1 + (-2.72 - 1.79i)T + (19.4 + 44.9i)T^{2} \) |
| 11 | \( 1 + (-1.72 - 14.7i)T + (-117. + 27.9i)T^{2} \) |
| 13 | \( 1 + (1.56 - 5.21i)T + (-141. - 92.8i)T^{2} \) |
| 17 | \( 1 + (6.21 - 1.09i)T + (271. - 98.8i)T^{2} \) |
| 19 | \( 1 + (-4.07 + 23.0i)T + (-339. - 123. i)T^{2} \) |
| 23 | \( 1 + (-6.78 - 10.3i)T + (-209. + 485. i)T^{2} \) |
| 29 | \( 1 + (-2.05 - 1.94i)T + (48.8 + 839. i)T^{2} \) |
| 31 | \( 1 + (0.593 + 10.1i)T + (-954. + 111. i)T^{2} \) |
| 37 | \( 1 + (-8.05 - 2.93i)T + (1.04e3 + 879. i)T^{2} \) |
| 41 | \( 1 + (-2.32 - 9.79i)T + (-1.50e3 + 754. i)T^{2} \) |
| 43 | \( 1 + (-19.9 + 46.3i)T + (-1.26e3 - 1.34e3i)T^{2} \) |
| 47 | \( 1 + (-63.1 - 3.67i)T + (2.19e3 + 256. i)T^{2} \) |
| 53 | \( 1 + (-51.7 + 29.8i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-1.44 + 12.4i)T + (-3.38e3 - 802. i)T^{2} \) |
| 61 | \( 1 + (-14.0 + 7.04i)T + (2.22e3 - 2.98e3i)T^{2} \) |
| 67 | \( 1 + (53.7 + 56.9i)T + (-261. + 4.48e3i)T^{2} \) |
| 71 | \( 1 + (86.4 + 103. i)T + (-875. + 4.96e3i)T^{2} \) |
| 73 | \( 1 + (-10.9 - 9.16i)T + (925. + 5.24e3i)T^{2} \) |
| 79 | \( 1 + (-121. - 28.9i)T + (5.57e3 + 2.80e3i)T^{2} \) |
| 83 | \( 1 + (1.04 - 4.40i)T + (-6.15e3 - 3.09e3i)T^{2} \) |
| 89 | \( 1 + (63.0 - 75.1i)T + (-1.37e3 - 7.80e3i)T^{2} \) |
| 97 | \( 1 + (3.82 + 5.14i)T + (-2.69e3 + 9.01e3i)T^{2} \) |
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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
−13.25083873266743504077758325383, −11.99166388898642595990616291230, −10.62260427429434236371995419208, −9.552677036019725229482882300900, −8.890565346730764824625830435723, −7.74237270145462484640942354443, −6.84054866756608928626892570400, −5.08751341203090813840979197974, −4.17989977065024226411335052982, −2.23829960266327011984736627951,
1.18569936891457201746805246821, 2.85429707514618698237255273006, 4.00620350201316321889928390404, 5.84433357634787874524697298261, 7.41753938520923936015007895416, 8.339997266694874690699422708308, 9.157786529893138836969490700135, 10.36869890741115210558977956156, 11.36745576360661593841142859821, 12.39103983297197382662521988196
