L(s) = 1 | + (−0.402 − 0.194i)2-s + (−1.12 − 1.40i)4-s + (0.162 + 0.710i)5-s + (−1.42 − 2.22i)7-s + (0.378 + 1.65i)8-s + (0.0725 − 0.317i)10-s + (−3.11 − 1.50i)11-s + (2.26 + 1.09i)13-s + (0.143 + 1.17i)14-s + (−0.631 + 2.76i)16-s + (−3.59 + 4.51i)17-s − 5.20·19-s + (0.817 − 1.02i)20-s + (0.965 + 1.21i)22-s + (−5.47 − 6.86i)23-s + ⋯ |
L(s) = 1 | + (−0.284 − 0.137i)2-s + (−0.561 − 0.703i)4-s + (0.0724 + 0.317i)5-s + (−0.539 − 0.841i)7-s + (0.133 + 0.585i)8-s + (0.0229 − 0.100i)10-s + (−0.940 − 0.452i)11-s + (0.628 + 0.302i)13-s + (0.0382 + 0.313i)14-s + (−0.157 + 0.692i)16-s + (−0.873 + 1.09i)17-s − 1.19·19-s + (0.182 − 0.229i)20-s + (0.205 + 0.258i)22-s + (−1.14 − 1.43i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0158242 + 0.222599i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0158242 + 0.222599i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (1.42 + 2.22i)T \) |
good | 2 | \( 1 + (0.402 + 0.194i)T + (1.24 + 1.56i)T^{2} \) |
| 5 | \( 1 + (-0.162 - 0.710i)T + (-4.50 + 2.16i)T^{2} \) |
| 11 | \( 1 + (3.11 + 1.50i)T + (6.85 + 8.60i)T^{2} \) |
| 13 | \( 1 + (-2.26 - 1.09i)T + (8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (3.59 - 4.51i)T + (-3.78 - 16.5i)T^{2} \) |
| 19 | \( 1 + 5.20T + 19T^{2} \) |
| 23 | \( 1 + (5.47 + 6.86i)T + (-5.11 + 22.4i)T^{2} \) |
| 29 | \( 1 + (2.54 - 3.19i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + 4.41T + 31T^{2} \) |
| 37 | \( 1 + (4.34 - 5.44i)T + (-8.23 - 36.0i)T^{2} \) |
| 41 | \( 1 + (-0.150 - 0.657i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (-1.08 + 4.76i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (-2.54 - 1.22i)T + (29.3 + 36.7i)T^{2} \) |
| 53 | \( 1 + (8.42 + 10.5i)T + (-11.7 + 51.6i)T^{2} \) |
| 59 | \( 1 + (0.598 - 2.62i)T + (-53.1 - 25.5i)T^{2} \) |
| 61 | \( 1 + (-8.78 + 11.0i)T + (-13.5 - 59.4i)T^{2} \) |
| 67 | \( 1 + 4.28T + 67T^{2} \) |
| 71 | \( 1 + (-2.84 - 3.56i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-0.515 + 0.248i)T + (45.5 - 57.0i)T^{2} \) |
| 79 | \( 1 - 4.07T + 79T^{2} \) |
| 83 | \( 1 + (-4.48 + 2.16i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-6.71 + 3.23i)T + (55.4 - 69.5i)T^{2} \) |
| 97 | \( 1 - 3.17T + 97T^{2} \) |
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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
−10.70105266583246736546208505566, −10.00530471812247267219022766477, −8.749002604683508714789079308509, −8.236378586886604307501482292133, −6.69110447127624059221111664243, −6.09524127691100145287462640233, −4.72719128225619834898513144412, −3.73183855113962563955591606307, −2.03472440730750278429613378143, −0.14671634969915827220656770724,
2.38630514892058925971253519428, 3.67716129619121146913640920857, 4.88544674591005798492041186274, 5.90190169576184340944391778116, 7.17944793072064548166274023582, 8.040610502091549090463952994733, 9.002667295184531662794779749728, 9.435169443179373871030718301567, 10.59878560621161790235751602593, 11.69442608028124496705974010929