| L(s) = 1 | + (−0.142 + 0.989i)2-s + (0.415 + 0.909i)3-s + (−0.959 − 0.281i)4-s + (−0.186 − 0.215i)5-s + (−0.959 + 0.281i)6-s + (−0.841 − 0.540i)7-s + (0.415 − 0.909i)8-s + (−0.654 + 0.755i)9-s + (0.239 − 0.153i)10-s + (0.181 + 1.26i)11-s + (−0.142 − 0.989i)12-s + (−3.53 + 2.27i)13-s + (0.654 − 0.755i)14-s + (0.118 − 0.258i)15-s + (0.841 + 0.540i)16-s + (−4.19 + 1.23i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 + 0.699i)2-s + (0.239 + 0.525i)3-s + (−0.479 − 0.140i)4-s + (−0.0833 − 0.0961i)5-s + (−0.391 + 0.115i)6-s + (−0.317 − 0.204i)7-s + (0.146 − 0.321i)8-s + (−0.218 + 0.251i)9-s + (0.0757 − 0.0486i)10-s + (0.0547 + 0.381i)11-s + (−0.0410 − 0.285i)12-s + (−0.980 + 0.630i)13-s + (0.175 − 0.201i)14-s + (0.0305 − 0.0668i)15-s + (0.210 + 0.135i)16-s + (−1.01 + 0.298i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.139202 - 0.372086i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.139202 - 0.372086i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 - 0.989i)T \) |
| 3 | \( 1 + (-0.415 - 0.909i)T \) |
| 7 | \( 1 + (0.841 + 0.540i)T \) |
| 23 | \( 1 + (3.51 + 3.26i)T \) |
| good | 5 | \( 1 + (0.186 + 0.215i)T + (-0.711 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.181 - 1.26i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (3.53 - 2.27i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (4.19 - 1.23i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-2.72 - 0.800i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (4.85 - 1.42i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (2.46 - 5.40i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (5.75 - 6.64i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-0.171 - 0.197i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (2.02 + 4.42i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 5.56T + 47T^{2} \) |
| 53 | \( 1 + (3.42 + 2.20i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (0.715 - 0.460i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-5.19 + 11.3i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-1.48 + 10.3i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (2.19 - 15.2i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-6.50 - 1.91i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-0.0890 + 0.0572i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (8.36 - 9.65i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-3.69 - 8.08i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-4.54 - 5.24i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−10.17605354691597647644044378045, −9.691110650062479389637400296103, −8.815540124204937970228939526366, −8.108313537045230736570201320493, −7.04704206663171799475817985996, −6.48996492470139092936197864430, −5.18965870194389658302825544808, −4.51696494436635917454617349929, −3.54054820482745576007878347567, −2.07577061479887687552155832368,
0.17290492100857588420422093534, 1.87223235692906430573780607049, 2.88027781857847215100131729434, 3.79346320449597376150883080491, 5.10941780157555933800097479146, 5.98483597611173397489110546592, 7.24534736777674584709571465642, 7.75247089487144070550506160132, 8.917184857945064722531872040138, 9.445815828917706623082737778057
