L(s) = 1 | + (−0.192 − 2.56i)2-s + (−4.56 + 0.687i)4-s + (−1.80 + 1.67i)5-s + (1.97 + 1.76i)7-s + (1.49 + 6.55i)8-s + (4.65 + 4.31i)10-s + (0.456 − 0.310i)11-s + (1.06 + 0.512i)13-s + (4.13 − 5.39i)14-s + (7.70 − 2.37i)16-s + (2.74 + 6.98i)17-s + (−0.499 − 0.865i)19-s + (7.09 − 8.90i)20-s + (−0.885 − 1.10i)22-s + (0.831 − 2.11i)23-s + ⋯ |
L(s) = 1 | + (−0.135 − 1.81i)2-s + (−2.28 + 0.343i)4-s + (−0.808 + 0.750i)5-s + (0.745 + 0.666i)7-s + (0.528 + 2.31i)8-s + (1.47 + 1.36i)10-s + (0.137 − 0.0937i)11-s + (0.295 + 0.142i)13-s + (1.10 − 1.44i)14-s + (1.92 − 0.594i)16-s + (0.665 + 1.69i)17-s + (−0.114 − 0.198i)19-s + (1.58 − 1.99i)20-s + (−0.188 − 0.236i)22-s + (0.173 − 0.441i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 + 0.601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.798 + 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.906985 - 0.303413i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.906985 - 0.303413i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-1.97 - 1.76i)T \) |
good | 2 | \( 1 + (0.192 + 2.56i)T + (-1.97 + 0.298i)T^{2} \) |
| 5 | \( 1 + (1.80 - 1.67i)T + (0.373 - 4.98i)T^{2} \) |
| 11 | \( 1 + (-0.456 + 0.310i)T + (4.01 - 10.2i)T^{2} \) |
| 13 | \( 1 + (-1.06 - 0.512i)T + (8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (-2.74 - 6.98i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (0.499 + 0.865i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.831 + 2.11i)T + (-16.8 - 15.6i)T^{2} \) |
| 29 | \( 1 + (-3.55 + 4.45i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (2.90 - 5.02i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.63 - 0.548i)T + (35.3 + 10.9i)T^{2} \) |
| 41 | \( 1 + (-0.206 - 0.902i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (0.949 - 4.15i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (-0.720 - 9.61i)T + (-46.4 + 7.00i)T^{2} \) |
| 53 | \( 1 + (7.73 - 1.16i)T + (50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (2.03 + 1.88i)T + (4.40 + 58.8i)T^{2} \) |
| 61 | \( 1 + (-7.05 - 1.06i)T + (58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (1.62 - 2.81i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.0286 + 0.0359i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-0.624 + 8.33i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (2.93 + 5.08i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-10.6 + 5.15i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-3.72 - 2.54i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 - 1.37T + 97T^{2} \) |
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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
−10.96914356370108488608251956981, −10.60776680734064231322485131221, −9.456251326840172726502183414597, −8.481265403136484830990387733065, −7.85225450675364094035454128926, −6.17808820241917816777118593689, −4.71348446072219445874237075070, −3.76248043499136803618925983847, −2.81188195208922876263801377236, −1.50772024083293694508071843115,
0.72327481674394730153985113087, 3.84443713111667246842942225906, 4.79939279596150242882483846627, 5.43863753382407293961708761917, 6.84086131068982491610351338519, 7.56579887024698358872130216003, 8.168739685947018253751164200169, 8.985862503599880949422790860479, 9.913543334988295256694626255239, 11.26542541527538758579308824671