| L(s) = 1 | + (−0.142 − 0.989i)2-s + (0.909 + 0.415i)3-s + (−0.959 + 0.281i)4-s + (−1.62 + 1.87i)5-s + (0.281 − 0.959i)6-s + (−2.24 − 1.39i)7-s + (0.415 + 0.909i)8-s + (0.654 + 0.755i)9-s + (2.08 + 1.34i)10-s + (6.16 + 0.886i)11-s + (−0.989 − 0.142i)12-s + (−0.803 + 1.25i)13-s + (−1.05 + 2.42i)14-s + (−2.25 + 1.02i)15-s + (0.841 − 0.540i)16-s + (−5.35 − 1.57i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (0.525 + 0.239i)3-s + (−0.479 + 0.140i)4-s + (−0.726 + 0.837i)5-s + (0.115 − 0.391i)6-s + (−0.850 − 0.526i)7-s + (0.146 + 0.321i)8-s + (0.218 + 0.251i)9-s + (0.659 + 0.423i)10-s + (1.85 + 0.267i)11-s + (−0.285 − 0.0410i)12-s + (−0.222 + 0.346i)13-s + (−0.282 + 0.648i)14-s + (−0.582 + 0.265i)15-s + (0.210 − 0.135i)16-s + (−1.29 − 0.381i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.363791 + 0.524384i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.363791 + 0.524384i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.909 - 0.415i)T \) |
| 7 | \( 1 + (2.24 + 1.39i)T \) |
| 23 | \( 1 + (4.71 - 0.879i)T \) |
| good | 5 | \( 1 + (1.62 - 1.87i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-6.16 - 0.886i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (0.803 - 1.25i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (5.35 + 1.57i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (4.29 - 1.26i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (3.68 + 1.08i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (4.34 - 1.98i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (6.36 - 5.51i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (0.544 + 0.472i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-8.55 - 3.90i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 13.2iT - 47T^{2} \) |
| 53 | \( 1 + (0.539 + 0.838i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-2.94 + 4.57i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-1.90 - 4.17i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (7.86 - 1.13i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (1.65 + 11.5i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (1.52 + 5.20i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (4.25 - 6.62i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-4.16 - 4.80i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-6.88 + 15.0i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (7.54 - 8.70i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.33075959856566266158301522918, −9.324195460303181522068670299572, −9.042453561381070291759657098451, −7.76484358541288934910141526933, −6.91846639482139640880810115000, −6.30175841642420213355073039167, −4.32262373577213071104136117842, −3.96493794513251797794175608635, −3.07634234772233125671693436693, −1.80665799498118375362116857431,
0.27888518092920031743016728025, 2.04765265556420893972574279808, 3.81331425219781391448065864166, 4.16230349816020696474506889729, 5.61988874201621878262203988697, 6.51041173672755857052273590918, 7.13677469520104509039475623082, 8.308872597935547670230469876703, 8.967584311919334469273091129670, 9.130789097421897933244302850624
